ACCESSIBILITY IN EUCLIDEAN N-SPACE
WITH APPLICATION TO DIFFERENTIABILITY THEOREMS
DISSERTATION
Presented in Partial Fulfillment of the Requirements
for the Degree Doctor of Philosophy in the
Graduate School of the Ohio State
University
By
ALBERT GEORGE FADELL, B.A., M.A.
The Ohio State University
19$h
Approved by:
'"’’V a W Y j o u M"
Adviser Acknowledgement
The writer is indebted to Professor T* Rado for his valuable suggestions in the preparation of this dissertation
» i — Table of Contents
Page
I. PRELIMINARIES
1.1, Euclidean n-space ...... • 1
1.2, Lebesgue Measure ...... 5
1.3, Sections ...... 1°
I.iu Lebesgue Density...... 21
1,5* Real-valued Functions ..••«••..••• 36
II, THE SECTIONAL DENSITY THEOREM...... kZ
in. THE ACCESSIBLE S E T ...... 52
17. THE RADEMACHER-STEPANOFF THEOREM...... 63
V. THE STEPANOFF T H E O R E M ...... 85
- 11 - INTRODUCTION
Frequently, theorems are stated and proved for the case of the
plane, and the validity of their n-dimensional analogs is asserted
without formal proof. Most often the n-dimensional analogs are
indeed valid, as asserted, and moreover admit of a proof differing
from the planar case only in notation. Nevertheless, there arise
planar theorems for which either (a) the n-dimensional analog is
false, or (b) the n-dimensional. analog is in fact true, but requires
a new method of proof. Situation (a) is best exemplified by an
outstanding theorem concerning the existence almost everywhere of
a regular approximate total differential (see Rado [7]: numbers in
[ ] refer to the bibliography at the end) which is true for the
plane but is known to fail for the higher dimensions* On the other
hand, situation (b) is brought out by the Rademacher-Stepanoff
Theorem (see IV.10) concerning the existence almost everywhere of
a total differential. For both Rademacher’s proof [6] of his
original restricted planar form of the theorem and Stepanoff's
proof [9 ] of the present complete planar form utilize crucially
a unique property of the plane not possessed by the higher
dimensions. Yet both Rademacher and Stepanoff, overlooking
this fact, assert that their proofs readily extend to the n- dimensional case. One purpose of this dissertation, therefore,
is to present in full a formal proof of the Rademacher-Stepanoff
Theorem for Euclidean n-space. To this end we have introduced the concept of accessibility (see 111*1 ), the study of which is the
subject of chapter III* The need for making £ -estimates along
lines parallel to the coordinate axes motivates the notion of
accessibility* Our fundamental theorem is the so-called
Accessibility Theorem (see 111*10), the proof of which requires
the self-important Sectional Density Theorem (see 11*18), the
latter theorem being the n-dimensional analog of the Linear Density
Theorem (see Saks [8], Chapter 9, Theorem 11.1)« With the aid of
the Accessibility Theorem as a basic tool, ve are able to prove
in a rather natural way the Rademacher-Stepanoff Theorem for
Euclidean n-space* Having thus proved the Rademacher-Stepanoff
Theorem we apply it together with the Accessibility Theorem to obtain a proof of the Stepanoff Theorem for Euclidean n-space
(see V*l;8 ), which characterizes the existence almost everywhere of an approximate total differential (see V.32)* We should remark that the proof of the planar Stepanoff Theorem (see Stepanoff [9] and Saks [8], chapter VII, Theorem 12.2) can be modified readily by an inductive process to take in the n-dimensional case.
Nevertheless, our approach yields in addition to the Stepanoff
Theorem other pertinent theorems of independent interest (for example, V.10, V.31, and V.39), and utilises in a rather natural way the already known Rademacher-Stepanoff Theorem* I. PRELIMINARIES
1.1 Euclidean Spaces
1.1.1. Notation. Given two arbitrary sets of elements S^ and S^
we write S , C S , , or S„”^ S , when every element of S. is an 1 2 2 1 1 element of S . When S C S and 3 C S , we write S * S • 2 12 2 1 12 Also, x e S means x is an element of the set S. By the empty
set we mean the set without any element, and we denote it by 0 .
We define S^ - to be the set consisting of all elements x
wliich belong to S^ but not to S^. If ^ is any finite or
infinite collection of sets we define the union iJ S to be the
set of all x contained in at least one of the sets S, and the
intersection O S to be the set of all x belonging to all of
the sets S. Given an arbitrary sequence of sets S^, i = 1,2,..., we denote by lim sup the set consisting of all points x which belong to S for infinitely many indices i. If Si<— • S , i. i+1 i - 1,2,,.. , we set li^i j and if ,
i » 1,2,..., we set lim S. » O s . • i x ± x
1.1.2. Continuation. For each positive integer n we denote
Euclidean n-space by Rn , whose elements (points) are ordered 1 n n-tuples (x ,..,x ) of real numbers. Usually we abbreviate by 1 n writing x for (x ,...,x )• A standing notational convention will be the following. If any letter, with or without affixes, is used to denote a point in RH the n numbers defining the point
-1- will be denoted by the same symbol (with the same affixes if any) with superscripts \ n . Thus, if we speak of pQ in Rn, we mean the ordered n-tuple ( p ^ , . , p n) « Moreover, for a given o o positive integer j < n, the symbol p^ denotes the j-ih coordinate C of the point p (identified by the context), o
n 1,1.3* Definition, Given two points p, q e R we define the distance
li P - q U from p to q by the formula
U p - q u - [ (p1 - q1)2 r i«l
1,1, U, Theorem, The distance function satisfies the following four properties:
(i) p - q tt > 0 for all pairs p,q s R*1 . n (ii) Given p,q e R , then p - q 4 « 0 if and only if p ■ q ,
(iii) |\ p - q U - \\ q - p U for all pairs p,q e r” •
(iv) Given p,q,r e Rn » there follows the "triangle inequality"
It p - q u ♦ itq - ru > lip - r n •
Proof: See McShane [§*]» section 2,
1.5. Definition. A point p e Rn is termed the limit of a sequence of points p.^ s Rn if ti P - P ^ U -^ 0, We write p » lim p^ , or p± - - P as i / • •
1 n 1.6. Theorem, A point p ■ (p »,..,P ) is the limit of a sequence of I n j j points p. » (p. »•••»? ) if ^ d only if p — ^ p as i jr « i 1 i for j " l,2,«..,n *
Proof: See Caratheodory [1], llh, Satz 5*
T.1.7* Definition. A point p is called an accumulation point of n a subset S ofF. if p is the limit of a infinite sequence of distinct points p e S • Then closure S is the union of S and the set of accumulation points of S, A set S is closed if S « closure 3; that is, S is closed if every point which is the limit of a sequence of points of S must belong to S. A set S is open if Rn-S is closed*
1*1*9. Theorem* The class of closed subsets of Rn is closed under the operations of finite unions and arbitrary intersections. The class of open subsets of Rn is closed under the operations of finite intersections and arbitrary unions. The closure of a set is a closed set.
Proof: See McShane C*J. section 2*
1.1*9. Definition. Given a finite sequence of real numbers a^, b_^, a^, b^ such thatl^ - a_^ -Ib^ - aji, i, j » l,...,n , the set ^ x \ a^ < < b^, i ■ l,...,n ^ is termed an open oriented n-eube. If all of the signs < are replaced by the sign < , the set is called a closed oriented n-cube* If all, some, or none of the signs < are replaced by < the set is called simply an oriented n-cube* The numberlb^ - a^l is called the side-1ength of the
-3- n-cube.
1*1*10, Theorem. If we define an isolated point of a set S to be n a point in R which is not an accumulation point of S, then the set
of points of S which are also isolated points of S is countable*
Proof: We merely sketch the proof. Note first that the class of
'•rational1' n-cubes Q » < x1 < | , where are required to be rational numbers, is countable and covers Rn. With each
isolated point p e S we associate a rational n-eube Q such that p
is the only point of S contained in Q. Since p is isolated, such
a Q exists. Thus the isolated points of S are in a one-to-one
correspondence with a subset of a countable set and hence themselves
constitute a countable set,
1,1,11, Theorem, Let T denote a one-to-one transformation (function) from an arbitrary set X onto an arbitrary set Y. Then the following formulas hold:
(i) T(CJs) - Vj TS,
(ii) T(f\s) - 'I t s ,
(iii) T ^ - S g ) - TS1 - TS? , where S , S 41 X and the unions and intersections are taken over 1 2 any non-empty class of subsets S of X,
Proof: See Kuratowski (3]> ^ 3, sections I and II,
-U- 1.2. Lebesgue Measure
1*2.1. For the following definitions and theorems of this section
1.2 the reader is referred to Saks [8], Chapter II,£U,|2 7>
Chapter III, JfJ 1 to jj 6, inclusive.
1.2*2, Theorem. If we denote by the n-dimensional Lebesgue outer measure (see Saks, Chapter II,£ it), then L° is defined on the class of all subsets of R11 and has the following properties:
(i) S is a non-negative real number or + ,
(ii) L° S < L° So whenever S..— S , n 1 - n ^ 1 2 (iii) L° ( \JS.) < i L° S. for all sequences S., i ■ 1,2,. n i-1 1 “ i-1 n 1 1
1*2.3. Definition. A subset S of Rn is termed L -measurable if for _u______every subset E of Rn we have the identity
L® (S = I E) ♦ L® I(Rn-S) f\E] - l£E .
T.2.it. Theorem. The class of L -measurable subsets of Rn is closed n under the operations of countable unions, countable intersections, and differences, and contains the empty set and the whole space
(see Saks, Chapter II, Theorem U.5).
1.2*5. Theorem. If we define as usual the class of Borel subsets Q n of R to be the smallest class of subsets of which (a) contains the closed and open subsets of Rn , and (b) is closed under the operations of countable unions, countable intersections, and differences, then the Borel subsets of Rn are Immeasurable (see
Saks, Chapter II, Theorem 7.U)•
1 .2.6. Theorem. Given a sequence of bounded Immeasurable subsets
of Rn , i • 1,2,..., and an arbitrary subset E of R11, we have the relations (see Saks, Chapter II, Theorem U.6).
(i) l£ [ li^i sup ( E H S ^ ) ] > ligi sup 1^ ( E H S^)
(ii) L° [ lim (E O Si)] - li| I°(E 0 S.), if either S . d S ^ , i * or S± i ** 1 >2 j««* #
1 . 2 . 7 . Theorem. Given an arbitrary subset S of Rn there exists a
Borel subset S* of Rn such that (see Saks, Chapter III, Theorem 6.8)
(i) S*-> S,
(ii) L°(S*i I E) • L ° ( S C\ F) for every Immeasurable subset E of
Rn.
The set S* is called a measurable hull for S.
1 . 2.8. Theorem. If ws denote by 1^ the restriction of 1^ to the class of Immeasurable subsets of Rn (that is, I^S is defined only when S is Immeasurable, and takes the value I ^ S ) , then Lr has the additive property (see Saks, Chapter II, Theorem 2**6) c o (i) I^( * 0 Sj ) » i Z H L 3. if SiOiS. - # for i / j, and hence i-1 i-1 11 J the subtractive property
(ii) y s ^ ) - - i*s2 , if s2c l Sl .
-6- 1*2.9* Theorem. Given any L -measurable subset S of Rn, there n exists a sequence of subsets of R , i « 1,2,..., for which the following statements hold (see Saks, Chapter III, Theorem 6«6(iv))
(i) Si is closed, i - 1,2,... ,
(ii) S± C S i+1< ^ S , i-1,2,... ,
(iii) I. (s - G s ) - 0 . i-1 1
1.2.10. Theorem. Given two L -measurable subsets S., of Rn, n 1 2 we have the formula
I (S, H S-) + I. (S ^ S j - L S ♦ L S . n 1 Z n 12 nl n2
Proofr We first express S ^ U as the disjoint union of Im measurable sets as follows:
s1U s 2 - (siO s 2)U[si-(sirvs2)]0[s2-(s1O s 2)) ,
Then by 1.2*8 we obtain
* v s!ns5) * V i - v sin y *
♦ L nS ? . V S 1OS2, , and so by conjoining terms the formula is proved*
1*2*11. Definition. Let S be an arbitrary subset of Rn. Then a property P is said to hold at (or for) almost every point of S if * there exists a subset S of S such that (ii) The property P holds at each point of S*.
1.2.12. Theorem. Let S be a bounded Ln-^neasurable subset of Rn , and let P denote a property defined at each point of S. Suppose that for any assigned € >■ 0 there exists an Immeasurable subset
E = E( 6 ) of S such that (a) Ln(S - E)-<6 , and (b) the proper ty P holds at almost every point of E. Then P holds at almost every point of S.
Proof: In accordance with the hypothesis select for each posi tive integer j a subset Ej of S such that
(1) Ln(S - Ej) *< 1 / 2 j ,
(2) P holds at almost every point of Ej .
Now by (2) (see 1.2.11) there exists for each j a subset E*j of Ej such that
(3) Ln(Ej - E * j ) = 0 ,
(4) P holds at each point of E*j .
Also, for each positive integer k we have the obvious inclu sions By (5), (3)» and (l) it follows that
q O Od o O (6) LAS — O E* ) i 2 ( S -E) -V 2 L ( E -E*) ^ ^ J21 y j*kv y j.-kv j j
-< 1/2 k_1
Noting that (6) holds for each k, we have that
LJ (7) Ln'S - jsl y “*1 j ) = 0 • 00 ll ^ But by (4 ) property P holds at each point of . . E and so J J with (7 ) the theorem follows.
-9- 1*3* Sections
1,3*1* Notation. For a given positive integer n » 2 let w ■
denote a proper subsequence of ^l,...,n^ , and * * i let it » j^,*..denote the complementary subsequence. Note
then that 1 < k < n-1 and once n is assigned the associated positive integer k is also assigned. We shall consider tt and v
X ti 21 as point functions in the following sense: if p ■ (p ,...,p) R , then
tip - (phl,...,phk) , «*p - (ph¥,...,ph*n*k ) •
Thus t t , tt* are functions defined on the space Rn and having their
images in the spaces R^, Rn'*k respectively. We call i*p the «• k ... projection of p into R , and tt p similarly*
n k 1*3*2* Continuation. Given a subset S of R and any point R # -l (see 1.3*3 below) we term the set w (S ^ v”Ax. ) the (w«xy) - section of S. Note that the set s O tr "Sc^. consists of points of S which have hj-th coordinates the same as the i-th coordinates of
Xjj,* Then v*(S f V n"*^rjc) is the aggregate of the ^-projections of these points, and therefore is a subset of R11”^. The reader will find it instructive to apply this notation in the case of the plane
(n - ?) *
1.3*3* Continuation* Given a positive integer k < n we will subscribe the k to a small Latin letter to denote a point of R^, and we will
- 10- superscribe the k to a capital Latin letter to denote subsets of
Rk. For example, ■€: Sk C_ . This notation will occur only
in this part 1.3 and in Chapter II, and is not to be confused with the usual notation for infinite sequences*
I*3«li» Continuation. Define for each positive integer 3, 2 < 3 < n
1 n Thus in accordance with 1.3*3, given p » (p ,...,p") we have
*jP ■ ■ (p1 j • •«jP*^**1) ♦
Also, given any subset S of Rn we will use the notation S(p^,...,pn) as defined by the formula
S(pj,*..,pn) - , if 2 < 1 < n * 3 3 3
Thus S(p^,*..,pn) denotes the set of points (a^,**.,x^) <=. such that
(x ,*..,x , p*^,»«»,p ) S *
Therefore the following formula is easily verified:
( S(pn ) ) (p^,*..,?1**1) - SCp*5,...,?11) if 2 < >J < n-1 .
The above notation will find application in Chapter III. But for
Chapter V, as well as for JJ1$. -, where line sections are needed we define the projections ita , ir* by the formulas where the symbol 3 3-s used to indicate the deletion or the singling out of the index i* Thus
/ I i+1 Ux * j W a p ■ (p »*«*»P » P ) > ®* P P • v 3 3 * The symbol p will always indicate p , that is, the j-th coordinate of p, and similarly for any snail Latin letter having j as a superscript*
1*3*?* Agreement* For the sequel of this part 1*3 let a subsequence i . * n of { l,...,n Y be thought of as fixed, and let s be the complementary* sub a equence.
T1 JC 1*3*6, Theorem* If C Sg <- R , then given any point p^^ R we have the inclusion
w " 2^ ) t ^ n ^ ) *
Proofs This follows at once from 1*3*1*
* v n»k 1*3*7* Theorem* Given i», « as in 1*3*1 let S , S be subsets of R^ and R0"^ respectively* Then
(i) «(«_1Ek 0 ir*"1E“-k ) . sk
(ii) .‘ ( n - y H V V * ) - a“-k .
Prooft We prove (i), and (ii) follows in a similar way* The inclusion (1) «(n"Jsk n » * " 1sn’k) c sk
- I k v is obvious, since nn S ■ S . Now assert that
(2) Sk C- 8 ( 1 1 ^ 0 s * " 1^ ) .
To prove (2) take any point (p*’,.*.,^) (q^,#..,qnF*k)ts s“-k • Then the point r * jr11) defined by the requirements hi i hi i r ■ p , i ■ l,...,k and r - q , i ■ l,...,n-k , 1 k lc & T v obviously has the properties wr«(p,...,p)tzS,«r « (q »»*«,qn ) B**k fc=. S . Thus (2) is true, and with (1) the theorem follows* o n i.3.c* Theorem. Given any sequence p ^ R •then p — p if w m and only if * * wp ■— *> »p , » PM —* * P # m n Proofx This follows at once from 1.1.6 . V 1.3.9. Theorem. Given any point p^ R the projection tr is a -1 k one-to-one mapping from n onto R • k it Proof: Let p^t: R be given. By 1.3.1 the projection * is single valued on Rn and hence on the subset *~^p^ of Rn. To show that ff* is univalent on tr"*^pjc, let p, q be two distinct points such that —1 # p,q€. » pjj , in other words, up * ffq * p^ • Then since » is complementary to w it is clear that it p and w q are distinct, -13- since otherwise the stated condition ffp ■ «q would require that p • q • To see that w is onto, take any point p ^ ^ (z. R * Then the point * p having v -coordinates the sane as the coordinates of p^jj, and u- coordinates the same as the coordinates of p^, surely has the properties wp • p^, s*p • P^jr • Thus, since the statement *p « p^ is equivalent to p t w"*^PkJ ^ e Projection a* is onto, and the theorem is proved# v 1*3*10* Theorem* Given any point p^tc R we have the following formulass (i) «*k u s> r\ - u**(sc\ *-\) , (ii) »*[( ^ S) (\ * " V k J " ^ H*(s ^ » (iii) «*C(sr s2) n *’^ 3 » **(s1 '\ 9m\ k ) - «*(s2r\ , where in (i) ,(ii) the 0 , fl are taken over any class of subsets S in Rn. Proofs Ve first note that by the distributive and associative lavs of sets we have the identities (1) ( Us ) n « U (S (\ w - ^ ) , (2) (fis) n »f\(sfi«*1pk) , o) (^ - s2)Pi e - 1^ - ( \ ^ « ~ \ ) - (s2 n « - \ k) . Now since by 1*3*9 the projection tt is one-to-one on p^, it follows from 1*1*11 that -lk- do »*[u (s n ."Sp j - j «*(sn«mlpk) , (5) **i n (s a - V'CSjO it*1^ ) - «*(S2n "“S^) , Now (1), (1*) Imply (i)j (2), (5) imply '(ii)} and (3),(6) imply (iii). Thus the lemma is proved* I*3*11* Theorem* Let S be a closed subset of Rn, and let there be lc given a sequence p ^ <= R , i ■ 1,2,..., with pj^ — =*■ as i — ^ lim sup » (S Pi w"*pj^ ) •— v (S \ • Proof t Take any point (1) P^jj <= lim sup if*(S 0 • Then (see 1*1*1 ) there is an infinite subsequence ^ of the positive integers i such that <2 > P n - k e ** In view of (2) there exists a sequence of points (3) S such that -15- U ) - p ^ , (*) » Pa-k • * But by hypothesis p j ^ > p^ » and bo Py-^ r * Therefore from (1;) we have (6) nqi^ ^ Pk • Consider now the point r ^ Rn defined uniquely (see 1*3*9 ) by the conditions (7) «*r - , (8) irr - pk * Relations ( 5 ) , (6) imply that (see 1*3.8 ) converges and indeed by (7)* (8) we have (9) r * • But since by hypothesis S is closed it follows from (9), (3) that (10) res* Now relations (7)j(8)»(10) require that (11) p <= A s H v ^ a ) n-k K Thus (1) implies (11), and so the theorem is proved* - 16- I *3• I t* Fubini Theorem. Let S be a bounded L -measurable subset n * 1 11 of R • Then the section « (S C\ i T Is ^-measurable for almost every point t t > the function L° , [«*(S \ ) ] is L « * n-k K n integrable, and si° t«*(s a ] d - i^s Rk Prooft The proof (which here we omit) nay be accomplished by applying the technique used by Saks [ 8 ] ^ 8, pp. 76-62 • 1.3*13. Corollary* Under the hypothesis of 1.3*12, I^S ■ 0 if and only if (S H a \ ) 1 - 0 for almost every point x ^ fe R * Proofr This follows at once from 1.3*12 • I.3*lU* Theorem* Let S be an Immeasurable subset of R®. Then for almost every point p € Rn the section «*(S f\ wT^Wp) is I ^ k ” measurable. Proofr Suppose, contrary to the theorem, that there exists a n subset E of R such that (1) L E > 0 , ' n ^ ]L (2) p t E implies that w (S C\»" wp) is not I^-measurable* Prom (1) and 1*3*13 there follows the existence of a subset of Rk - 1 7 - such that (3) ^ ^ > 0 (h) E^ implies that L^^[«*(E Oiw^p^)] » 0 * Let there now be given a point k (£) P t E * k From (U) we know that » (E \ w p^) f fjf and so there exists a point (6) p fc E such that (7) «p - p • k Now (6) and (2) imply that w*(S n \p) is not I^^-aeasur^le, and ®o hy (7) we have that * **3' (8) «w ( s f U p^) is not L^^-neastcrable • Thus (5) implies (8), which by the Fubini Theorem (see 1*3*12) is impossible in view of (3)* Hence the existence of a set E satisfying (1),(2) is impossible, and so the theorem is proved* 1*3*15* Theorem* Let S be a subset of H11. Then given any point p <=. Rk the section A s ^ w " 3^ ) is a Borel, closed, or open subset of Rn"k according as S is a Borel, closed, or open subset of Rn* - 18- Proof t If S is closed^ the section tr*(sO can be easily shown to be c]dosed using 1.3*8* If S is open, then Rn-S is closed) «nd by 1*3.10 s*[(Rn-S)n * ~ \ k ] * A r 11^ - A s n ) - - R ^ - s*(S \ « - \ ) , * *1 and so n (S 11 t> p ) is open* For the case of S being a Borel set k * n ve consider the class B of those Borel subsets of R having the ^ Ip property that « (S f\ n~ p^) is a Borel subset of Rn” • By 1*3*10 ** and 1*2*6 the class B is closed under countable unions and differences, and (as we have just shown) contains the closed and open sets. Now B being a subclass of the class of Borel subsets n of R , the theorem follows fdtwri 1*2.5 • 1.3*16* Definition. Let Sk , S1** be subsets of Rk, R0"* respectively* We define the Cartesian product Sk X s°“k to be a subset of R11 defined by the fommla Sk X s““k • ^p |p - (Pj^P^fc) » Pk *= Sk , P^j, vs Sn“k i . 1*3*17* Theorem* If S^, are respectively I y measurable> Jf y. If measurable sets then S X S is Lyineasurable and Ln(Sk x S®“k) - (I^Sk) . (ik.kS®’*) . Proofs See MacShane [5s], Theorem 25*7, p* lii5 * -19- k 1.5.18 Theorem. The set S X S is a Borel. closed, or open subset of R11 acoording as Sk , Sn”k are Borel, closed, or open subsets of Rk , R11-^ respectirely. Proof* See McShane [5a], Theorem 25.7 (in the proof), p. 145 for the fact that Sk /c Sn ”k is a d o s e d or open subset of B?1 according k k _n-k as S , S are closed or open subsets of R , K respectirely. k & . Consider now the class B of the Borel subsets of R . Let B denote the subclass of B defined by the property that Sk € B* implies that Sk a R^"-^ is a Borel subset of R11 . Observe that B* is closed under the operations of countable unions, countable intersections, and differences « by reason of 1.2. 4 and the following easily verified formulas: (i) (U sM x h " - ^ U ( S k ^ R a-k ), (ii) ( O s k ) = f\ (sk x ), (iii) (Sk - s £ ) « R n”k = (Sk X &-*) - (s| x ) . Thus, sinoe by our initial statement B* contains the open and closed subsets of R^", we hare that B = B* - that is, SkxRP“k is a Borel subset of R“ if Sk is a Borel subset of Rk. By a similar argument I ^ x S n"k is a Borel subset of R11 if Sn_k is a Borel subset of R?1”^. Accordingly, in view of 1.2.4 and the obvious formula Sk X Sn *k - (sk < R11^ ) H (Rk x sn *k ) we have that SkASn”k is a Borel subset of R11 if S^, Sn “k are Borel subsets of R^, Rn_lc respectively. Thus the theorem it proved. -20- I*U Lebesgue Density n I*l;*l« Definition* Let S be an arbitrary subset of R , and let a point n p t R be given. Then p is termed a point of outer density for S if the following condition is satisfied; for any assigned tr > 0 there exists a £ - S(€ ) > 0 such that for any oriented n-cube Q of side-length s the inequality (sf\ Q) > (1 - u) sn holds if p w Q and a < 5 * , n I,U,2* Notation# We denote by *_XS the set of points of R which are points of outer density for S# n I*lu3* Lebesgue Density Theorem* Given an arbitrary subset S of R , we have the relation L° (S -z_^S) - 0 . n Proof: See Saks [ 8] , Theorem 10.^ page 129 * I» k ,h * Theorem* Given S-C. S <— R11 • we have the inclusion / S s J - / ^ S « 1 2 1 2 Proof: This follows at once from I*lwl, since L° is monotone* n I*U*5* Theorem: Given any L -measurable subsets S.., S of Rn we have n «*• z the formula - 21- Proof: The inclusion (1) sg) c. A s 1 rA A s 2 follows at once since A is monotone (see I*iuU) • Assert now that (2) a S j ' l *->Sg A ( s 1 ^ S? ) * To prove (2) take any point of p t ^ such that (3) p ^ , p c= ^S2 , and assign any <=. > 0 • Then (see I*U*I ) there exist, numbers 5- , ^ such that* for 1 2 any n-cube Q of side-length s, if p Q there follow the inequalities (U) 1 (S-Pl Q) > (L.( t/2) ) sn if s < 5 n 1 -»• (5) Ln (S2 f\ Q) > Define S - *in 5., ) , and let there be assigned an n-cube Q of side length s < c5 • Then Inequalities (10 and (5) hold* But application of 1*2*10 to the pair Q , S j O Q yields Sg 1 q) ♦ y t ^ a q) u (s2a ♦ Ife (S2 ^ Q) , -22- and so Ln(Sin Q) - Q) + ^ Q) * V * Using (li) and (5) we obtain L (S. i S 1 Q) > (1 - ( c/2) ) sc ♦ (1 - U / 2 ) ) s* - en - n -l ■ ( 1 - o)sn • Accordingly, (see I.U.l) since € was arbitrary,we have (6) P 6 ^ ( s 1 n s 2), and so since thus (3) iaplies (6) assertion (2) is true* Now (1) and (2) complete the proof* I«lu6. Lenina. Let Q be any closed oriented n-cube of side-lengtb s. Then there exists a sequence of n-cubes Q^, k * 1,2,..., for which the following statements hold: (i) is open, (i d (in) a - ii= u k - co , (iv) s ■ li» s^ as k ^ . Proof: Let Q - ^ x I ^ < a?" < jj Define for each positive integer k the n-cube \ • N^x I a± - (1A) < x1 < \ + (1A) | -23- Then clearly each is open and (see I.1.1 ) Q * lim ^ , as k -—f * Also since Q has side-length s - bA - a. and Q. has side-1 ength s^ ** b i * *1 + (2/k) it is clear that ■?s as k -? c-xj * Thus the lemma is proved* . n n 1*4*7* Theorem* Let S be any subset of R , and let p R • Then p S (see I*lu2) if and only if for any assigned ^ > G there exists a > » ^ ) > 0 such that given any open oriented n-cube Q of side-length s the statements -5 < and p Q Imply L® (S \ Q) > (1 . fc)sn * Proof: The necessity follows at once from the definition I.lul since the class of open oriented n-cubes is a subclass of the class of all oriented n-cubes. To prove the sufficiency} let c > 0 be assigned. Then by hypothesis there exists a .5” ■ ^"( t ) > 0 such that given an open oriented n-cube Q of side-length s we have (1) ij (S \ Q) > (1* c/2)sn if s < 5 , pc- Q . Now take any oriented n-cube Q of side-length s, and assume that (3) a < S, (J0 pcQ* — 2U— Let Q denote closure Q, whence (see 1.1.8) Q is closed. In accor dance with 1.4.6 let t k=l,2,..., be a sequence of oriented n-cubes of side-length sk having the following properties (5) - (8): (5) Qk is open, (6) 0 Q 13 Q , k 1 * (7) Q — lim Qk as k 0 0 , (8) S =r lim s^ as k o o . By (4 ), (6) we have that p 6. for each k ; and (3), (8) imply that sk -C S for k large enough, say k i kQ . Accordingly, in view of (5) it follows that (9) ^°(S ^ ^k^ ^ (l “ (^/2)) sn for k "Z. kQ . k Now, inasmuch as the sets form a descending sequence of Ln-meas- urable sets with limit T , we may pass to the limit in (9) (in view of 1.2.6) and obtain (see (8)) (10) L°(S O Q) i (1 - (6/2)) sn < (1 - e ) s11. Since the boundaryof n-cubes is easily shown to have ^-measure zero, the relation s HQ - ( s D Q) U £s 0( Q - Q)] yields (11) L°(sOQ) £ L°(S0 Q) + L°[s f|(Q - Q)| = L0(sfU) . n n n »• j n But the inclusion Q d Q requires (12) L°(S f\ Q) ■£= L°(sfI Q) . n n - 25 - Since XH).tlad (12) Imply* L°(S H $) - L°(S \ Q) , (30) yields n a (1 3 ) l (s i i q) > (i - e)sn , n and so since (3), (U) imply (13), application of I*i*,l requires p t A S , Accordingly, the theorem is proved, I,fc,8* Theorem, Given n > 2 let S be a I^j-neasurable subset S"1 i n. n of R and let p * (P'S***? ) <=> Bt such that * 1 n~l *np ** (p *,**»P ) *= S Then (see 1,3,16 ) p v_- (S*1"1 K R1 ) t ^ Tla»X Proof t Let > 0 be assigned. By 1,1;, 7 * cAS implies the existence of a number "S ■ S( o) > 0 such that given any open iwl oriented (n-l)-cube Q of side-length s ve have the inequality (1) 1 . (s”-1 ! cT 1) > (1 - J S - 1 if . < Y , »*p <= Q1^ 1 . n»i n Let (2) Q • ^ x \ a^ < at* < ^ be an open oriented n-cube side-length s , (3) p c Q , a < S , Then (see 1,3*4 for notation) by (2) it is clear (since a^ < pn < bn) that the section -26- (U) QCp 0) is an open oriented (n»l)-cube of side-length s, and by (3) ve note that (5) «Jp <=. fl(p“) , Then, in view of (3)> (h) , (5), it follows from (1) that (6) ^ [ S 1*-1 * QCP11)] > fl - O s * " 1 . But dearly we have the identity (7) [S^fiQCpP) } X I - ( S ^ X R1) ^ Q where I is the open interval of real numbers ^ with an < < bn « n-1 Noting (6), (7) and the hypothesis that S is I^j-measurable, we apply 1*3* 17 and thus obtain (8) Ita[(Sn “1 X R1) '1 Q ] - ys*-1 ] * % I > > (1 - Cr)a0-1. S ■ (1 •<= )sn • Briefly, (8) says (9) y (S®"1* R1) fI Q] > (1- (=)sn * Accordingly, since thus (2), (3) Imply (9) , it follows from I*li*7 that p & A ( s “J,x a1) , and so the theorem is proved* n I.lu9* Definition. Given any point p d H and any real number r > 0 we set K(pjr) ■ ( x i x t Rn , \\ p - x tt < r^. Then K(pjr) is termed the closed spherical neighborhood of p with radius r . I.U.10. Lemma. Every closed spherical neighborhood K(pjr) of a n point p is a closed (hence Immeasurable) subset of R , and I^KCpjr) - c rn , where c is a constant depending only upon n . Prooft See Mayrhofer [4 ], pp. 119-122. I.lull. Theorem. Let S be any subset of Rn, and let p t A S , Then for any assigned > 0 there exists a 5" - £>(<=)> 0 such that given any closed spherical neighborhood K(pjr) of p we have the inequality l£[S r \K(pjr)) > (1- e) LK(p;r) if 0 < r < 5 “ . Proof: Let c be the constant in the formula I^K(p,r) * cr* » and denote by Q(p,s) the closed oriented n-cube with center p and of side-length s . By hypothesis (1) p c a s , -28- Let t > 0 be any assigned nuifcer* Then by (1) and there exists a ■ S*( ) > 0 such that (2) l£ [ S U Q ( p j s ) ] > (1 - c2*,U ) a11 if s < 3 • Define (3) 5 - c* /2 , and take any ntufcer r with (1*) 0 < r < J . Let <3(p,s) be the (unique) closed oriented n-cube defined by s « 2r , and note that (3) ,(10 imply (5) s < S * • From (5), (2) it follows that (6) Lf[snQ(p,s)J > (1 - c2"°^ ) sn . n For brevity let K - K(pjr) , Q * Q(pjs) • Then in view of the obvious- inclusions S P i Q C (s n K) J (Q - K) ,KCQ, and the fact that Q,K are Ln<-seasurable, we have (7) l £(s a q ) < l® (s n k ) ♦ y q . k ) • • l£ (S f\ K) ♦ I^Q - y - -29- - l£ (sf\K) ♦ 2V - c r“ Applying (7) and (6) we obtain (recall a » 2r ) (8) I g ( s a K) > t®(S f\ > (1 - c2”n*i ) 2”^ - 2nrn ♦ er11 - - (1 - t) crn - (1 - K • Accordingly) since (U) implies (8), the theorem is proved* 1*1^12* Lemma* Let S, K^, Kg be subsets of R® , and let (i) K p Kg are L^-measurable j (ii) j (ill) L K g > j (iv) I§ (S Vi 1 ^) > (1 -**) 1 ^ . Then L® (S H Kg) > 0 Proof: Suppose, contrary to the lemma, that (1) Ln (sf U 2) - 0 B y (i), (ii) we have (see 1*2*8 ) (8) V i * - *h<*rt> • But in view of the obvious Identity S P\ (t^Kg) - S f 4 - s ' \ Kg we obtain the inequality (3) l£[S f \ (Ki-Kg) ] > l£(S rk Kj) - L°(S \ Kg) . But because of (1) it follows from (3) that (U) l£[S i A (I^-Kg)] - L°(S ,1^) • Application of (U), (ir),(iii) yields (5) I?[S (K^)]>(l-t)L°K1 - LnK1 - 2 Vi ' Vi • Now it follows from (2) and (5>) that (6) - ycs - > yts A (y*,)l » -Vi~ Vt 4 But (6) is Inpossible, and so (1) cannot hold* According!/, the leraa follows* n I*JLi*13* Lemma* let p, q fc R , and let ^ be any assigned number with 0 < ^ < 1 • Then if we set h * U p - q i| the following statements hold (see I.U.9 for notation) t (i) K(qj i^h) CL. K(pj (1+^) h ) j (ii) L K(qj Ah ) > y? 2*n 1 K(pj (!♦ >!) *» ) ♦ n • n -31- Proof: To prove (i) we take any point x t K(qj ^ h) » and note that therefore by I.U.9 we have U q - x il < *\ h • Then using the triangle inequality we obtain tip - x H < ftp - qil ♦ ttq - xn < h + ^ h - (1 + t) h , and so indeed x ^ K(pj (l-*^)h ) • Thus (i) holds. To prove (ii) we apply I.Jj.10 and the bound ^ < 1 to obtain LK(q; ,^h) - c /“hn - “ 2-n c 2“hn > ® 2"n c(l+ ^)”hn - - ^(pjd+.^h ) , and so (ii) holds and the lemma follows* n I.li.lli. Theorem* Let S be any subset of R , and let p <-r S • Then for any assigned number ^ > 0 there exists a T « 5(^ ) > 0 such that to each point q t r f 1 with kip - qu < 3 there corresponds a point q^u: S satisfying the inequality UP - qU . Proof: Clearly we may assume without loss that < 1 • Then since by hypothesis p q iis there exists (see 1 *14*1 1 ) a £> » e: ( ^) > 0 such that for any closed spherical neighborhood K(p;r) of p with radius r > 0 we have - 32- (1) Ig [S H K(pjr)] > (1- I^K(p|r) if r Define a nnriber 5 by the equation (2 ) (1 ♦ ^ - S* Now take any point (3) q ^ with p - q tl < 5 • Upon setting (U) h - i[P - q u we note that (see I«U«13) since »j < 1 we hare the relations (5) K(q; -Vh) - K ( p j ( l ^ ) h ) , (6) LK(qj ^h) > °2’n D/CpjCl* -\) h) . But (2), (3)> (U) imply that (levph < S * , so that from (1) it follows that (7) l£[s OK(p;(l+'»\)h] > (1 - ^ “2-n) I^(pjO>*\)h) . Now let us put K^ • K(p;(l+V|)h), Kg ■ K(qj v^h) , c m ^ 2*n • Then noting (5), (6),(7) we may apply 1*1^12 to obtain I^£S< \K(qj^h)] > 0 and so there exists a point (8) S ( \ K(q; v^h) • But (8) implies that (9) \ u S and (with (10 ) that (10) W % - < -\h • ^ ii P - q .1 • Thus, (3) implies (9) and (10), and so the theorem improved. I.U*l5. Theorem. For fixed n > 1 there exists a number °\ (depending only on n) such that 0 < 4 < 1 and given any two points p^, pg tr we have (see I*lt«9 ) « V i iCPj* UPj-Pg** )f\ X(P2> fePj-Pgd ) 3 V*(Pi» aPi-Pgkl) + V ^ * ui^-Pgii) Proof* The proof (which we omit) follows essentially from the fact (see Mayrhofer (4 ], ^ 19) that under a linear transformation the L -measure of a set is multiplied by the determinant of the n transformation* I«lt*l6* Theorem* Let S^,Sg be two L^-measurable subsets of Rn , and let p^, Pg ^ R*1 • P®* (®ee 1*^*9 ) (i) - Ktpp Ul^-PgU ) , Kg » . *C P2, tt P^PgU ) $ (ii) -3U- and suppose (iii) y s ^ A > ( i - u ) ^ , y s 2 i \ k2) > (1 -«<) LbK2 • Then there exists a point p i Sg such that d P - P^n < 1 Pi - P2 it » d P - P2 li SwPi “ • Proof: Let (1 ) ®x-sx('Ki ,E2 -s2r^x2 . Then we have (see 1.2*10 ) (2) A e2) ♦ y o Eg) - 1^ ♦ 2y , (3) y ^ \ Kg) ♦ Ln(Kx j Kg) - y x ♦ y 2 , and (since E jl J Eg w ^ J Kg) (H) y ^ jig) < y K x j K g ). Successive application of (2), (iii), (U),(3)» and (ii) yields \(\(\ E2) - y 2 - y Ex J Eg) > > ( i - c O y x ♦ ( i ^ ( A ) y 2 - y i ^ j K g ) - - CL-oOCyyV 2) ’ I W 1V[2‘i“(!ci 1'1 V ] * - I* ( K ^ Kg) - c x f y ^ ♦ L^Kg) - 0 Thus E^P\ Eg^t and so by (1) and (i) the lemma follows* I.Uol7« Corollary. Let S,, S be two L,-measurable subsets of the 1 2 l -35- real-line R^, and let P^,P2 ^ 4et ^2 ^eno^e ^'wo intervals of length s= 2 |\ px - p2 |l with centers at p p P2 respectively, and suppose l a ^ O I])^(3/4)S , \ ( s 2 C\ i2) > ( 3 / 4 ) s . Then there exists a point p^ £ Sj n s2 such that 1 p^-p#\ — 1 P1 “ p2 1 , 1 P2 - P*\ |Pl - P2 I . Proof: Assuming without loss that I-., I are closed, we apply 1.4.16 with n s 1, I 5 K , I = K , and o( - 1/4. 1 1 2 2 1.5 Real-valued Functions on Rn 1.5.1 Agreement. For the following definitions and theorems let f be a given real-valued function defined on a given bounded subset S of Rn. (We shall understand the term real-valued to mean finite real-valued.) 1.5.2 Definition. Given a point p £ S, f 3s said to be continuous at p if lim f(x) exists and is equal to f(p) as x approaches p through S. If f is continuous at each point of S, then f is said to be continuous on S. 1.5.3 Theorem. If S is closed and f is continuous on S, then for every number c the set ^ P | P ^ S, f(p)c | is closed. -36- Proof: Given any sequence of points p. e S with >PQ and f(Pi) < c we have pQ t S since S is closed, and f(pQ) * lim f(Pi) < c since f is continuous at pQ. Thus the theorem follows* I*5.it* Theorem. If S is closed and f is continuous on S, then f is uniformly continuous on S, in the sense that for any assigned <= > 0 there exists a *''(-)> 0 such that If(x^) - whenever x , x c S and H r • x il < r « 1 2 l 2 Proofs See Graves [2], Chapter IV, Theorem 23* 1.5*5* Theorem. The stun, difference, and product of two continuous functions on S are continuous functions on S. Also, the absolute value of a continuous function on S is a continuous function on S« Proof: See Graves [2], Chapter IV, Theorem 16. n 1*5*6* Definition. Given that S is an L -measurable subset of R , n then f is termed an I^-measurable function if for every number c the set ^ p \ f (p) < c is I^-raeasurable* 1.5*7* Theorem. The sum, difference, and product of two Im measurable functions on S are I^-measurable functions on S* The absolute value of an Immeasurable function on S is an Im measurable function on S. Proofx See Saks [8], Chapter 8, Theorems 8.2, 8.3» and 8*1;* 1.5.8. Lusin's Theorem. If S is an I^-measurable suboet of Rn and f is an Immeasurable function on S, then for any assigned G ■>■ 0 there exists a set E ■ E( G ) for which the following statements hold: (i) E d S, (ii) E is closed, (iii) Ln(S-E) < (iv) f is continuous on E * Proof: See Saks [8], Chapter III, Theorem 7.3* I*5*9. Corollary. Under the hypotheses of 1.5.8, assume that a property P holds at almost every point of S(see 1.2.11). Then for any assigned (i) E C S , (ii) E is closed, (iii) y s - E ) < , (iv) f is continuous on E, (iv) P holds at each point of E • Proof: By hypothesis there exists a subset S of S such that I (S-S*) « 0 and property P holds at eachpoint of S*• Let n € > 0 be assigned. Then applying Lusin's Theorem to the set S* we obtain a closed subset E of S* such that L^S^-E) < -38- Thus properties (i) - (v) are clearly satisfied, and so the corollary is proved* 1,5.30. Definition* Let a positive integer j < n be assigned. Then f is said to have an S-relative j-partial dexrivative at a point p c S if there exists a (finite) number A such that lim exists and equals A. We write f (p) * A 1*5.31. Theorem. If S is an I. •measurable subset of Rn and f n is continuous on S, then for each positive integer j < n the S-relative j-partial derivative f is an Immeasurable function on the subset of S consisting of all points p for which f^(p) exists. Proof: See Caratheodory [1], ^ 557, Satz 1 • 1*5.12. Definition, f is termed Lipschitzian on S if there exists a (finite) constant M > 0 such that the inequality IfCxj) - f(x2) \ < M U x1 - x2 U holds whenever x^, 1.5.13. Theorem. If f is Lipschitzian on S, then f is continuous on S. Proof: If lf(p) - f(x) \ < M l\p - x U , then I f(p) - f(x) i < whenever 11 p - x ii < and so the continuity of f on S is immediate* I*5*111* Theorem* If f is Idpschitzian on a finite interval [a,b] of the real line, then f is of bounded variation on I, in the sense that there exists a (finite) constant N such that for any sequence of non-overlapping intervals ^ [a,, b ^ J with end points in [a,b] we have | fCb^ - f(ai) \ < N . Proof: By hypothesis there exists a constant M such that I f(x^) - f(x?) V < K I x] - x? \ whenever x,, x„ CT fa,fc] . Then given any sequence of non-overlapping intervals ^ ^ai’^i^ ^ with end-points in [a,b] we have Z i I f(bt ) - f ( ^ ) [ < K 2 I bj- a1 1 £ M 1 b • a \ . Thus, if we set N • M \ b-a\, the theorem follows* 1*5*35* Theorem. If f is Lipschitaian on an n-cube Q, then for t each positive integer j < n the j-partial derivative f exists at J almost every point of Q* Proof: Since f is Lipschitaian on Q it is lipschitaian and therefore (by 1*5.35) of bounded variation with respect to each variable. The theorem therefore is a corollary of Caratheodory [1], ^ 559 > Satz 3* I*5*l6* Definition* Given any subset E / 0 and any point p e E , f is said to have an approximate E-relatlve limit at £ if there exists a (finite) number A such that for any assigned € > 0 (see 1 *14*2) we have p We write ap lim f(x) ■ A xeE, x ~?p 1*5*17* Theorem* If for a givensubset E of S, f has an approximate E-relative limit at p e E, then f has an approximate S-relative limit at p* Proof: This follows at once from I*5*l6 and I*li*ii* 1*5*18* Theorem* If p c E fl A E and f has an ordinary E-relative limit at p, then f has an approximate E-relative limit at p, and lim f(x) • ap lim f(x) * xeE,x-^p xeE, x-^p Proof: See Saks [8], Chapter VII, £ 3, remark 3*2* II. THE SECTIONAL DENSITY THEOREM 11.1. Definition# Let ^h^, ... » ^ be any proper subsequence of ^ | and let it* be the conpLelementary subsequence (see 1*3. I )• Then for each subset S of B® we define a subset D(ff,S) of BP as follows (see I«lu 2 ): D(i»,S) » ^ p \ p 6 R“ , flt*p ^ «*(S f\ w " 1!*?)}. 1 1 . 2 . Lemma. Under the hypotheses and notation of I I . l we have the formula D(w,S) - Vj ^[w*~1 A. «*(S C\ nC^Xjj) ] C\ xk where the union is taken over all points Proof* Let p €. D(«,S)• Then by I I . l we have that rf1 and w*p ^ A * * ( S f\ u^^iip). Then if we put p^ ■ up it is clear that P 6 « * ( S (\ w * 3^ ) ] 0 u*3^ , P j ^ e Rk , so that since p was arbitrary we have the inclusion (1) D(u,S) C_ U w*(S O W 3^)] H w"3^ # *k v Now suppose for some p ^ t E we have p «*(S H w“,1pjc) ] 0 w*3^ Then clearly p <=. tf1 > «p • > and -1*2- n*p (a Av*(S n n^p) , Accordingly (see 11*1) , p ^ D(*,S), and so (2) D(w,S) U [w*-1A **(S ( \ «-ax )] Pi ♦ *k With (1),(2) the lemma is proved* 11*3* Lemma . If r “, then C- ©(w,^) * Prooft This follows at once from 11*1, since ** and A are monotone (see 1*3* ^ and I*.U,4 )• II.Ii* Agreement* For paragraphs 11*5 to H.ll* inclusive assume that n * •••» 1 is a prop®? subsequence of ^1, • n| and S is a given subset of H?. Accordingly, for brevity we agree to write D instead of the more complete notation D(«,S)# It *5* Lemma. Given any point p^ Cs , then «*(D (1 m A w*(S 0 Proof* We first note that since « is single-valued, if ^ p^ we have * 0 $ so (see 11*2) ® ^ *"lpk “ 'l ^ [ n ^ A **(S ft ff"^) ] H w-1^ . \ 0 s " ^ • l x k • U [ w ^ A w*(S a u \ ) ] a * \ [ \ nm\ - xk » a «*(s aw"^) ] ft * —1*3— We now apply 1*3*7 (where we put » A**(sO^ ' - V v and thus obtain \[■*j a«*(sa < r \ ) ] a ^ - » Aw^sHe?*2^) * whence the lemma is proved* u 11*6* Lemma* Qiven any point p ^ G R , we have w*[(s-j5) n n"2^ ] * As 0 *r\) - A n H A\) Proof; Apply 1*3*10 with ■ S, Sg • D * II*?* Lemma* Given any point p^<= , we have ^w k \«*f(S-D)n«*Sk] \ - 0 Proof; By 11*5 we have the formula (1) **(D H w*2^ ) « Aw*(S O w ^ ) * From (1) and the Density Theorem (see I*U*3 ) there follows the equalities (2) C\ w*1^ ) • «*(D a w- 3 ^ ) ] - • r\ «*,1pk) • a . w*(s r\ * o « Now 11*6 , (2) imply Ln-k \ ^ trlpk3 \ * 0 » -UU- and so the lemma Is proved* 11*8* Lemma. If S*D is I^«aeasttrable, then yS-D) - 0 • Proof: Since by hypothesis S-D is Immeasurable, the lemma follows at once from II,7 and the Fubini Theorem (see 1*3*13 ) 11*9* Definition* For each pair of positive integers i, j we define a subset of Rn, depending upon « and S, as follows: p & D ^ if xwSc and only if pfc rf* and for any open oriented (n»k)-cube Q of side* length s the inequality holds whenever s < 1 and «*p <= Q53"^ * 3 11*10* Lenaa* In the notation of 11,9, we have D « ' • O D. i - i 1 J Proof: Assert first that o a (1) D C (I U i-1 J-l J To prove (1) take any point (2) p 6 D • Then by II.l (3) p C - J 11, -4*5- (It) «*p € A**(s 0 witfp) * Assign an arbitrary positive integer 1* Then (It) and the definition of outer density (see I*lt*l ) require the existence of a positive integer 3 such that (5) for any (hence any open) oriented (n-k)-cube of side»length •i »» Hwk s the statements s < =■ and u p c Q imply L°n.ktQn-kr\ «*(s a > (i-(i/ijsD"k. From (3), (5) and 11*9 it is clear that for the assigned i and the corresponding j we have p £= , and so since i was arbitrary it follows that O O o o (6) p c n u d i-i j-i 13 Mow since (2) implies (6), assertion (1) is true* Assert now that o o orO (7) n U D-. d - D * i-1 3-1 To prove (7) take any point O O ck , (8) p H U • i-1 j-l J Froa (8) it is clear that (9) p^R® , and (10) for any positive integer i there corresponds a positive integer 3 such that p -U6- In other terras (see H*9), (10) means that (11) for any positive integer i there corresponds a positive integer j such that given any open oriented (n-k)-cube Q of side-length a ve have the inequality L ' W a " ’* > "*(s n** \ i > ) ] »d-ci/i))»""k , if 8 < l/j and **p (z Qn"lc But in the characterization of points of outer density in terns of open oriented n-cubes (see I*!** 7 ) statement (11) implies that (12) n*p £ A «*(S H w-^p) * Now (9) and (12) require (see H*l) that (13) p <= D . Since thus (13) follows from (8), assertion (7) is true* With (1) and (7) the proof is complete* 11*11* Lemma* If S is closed, then for each pair of positiveintegers i,j the set of 11*9 is also closed* Proof: Let a pair of positive integers i,j be held fixed*Let p t t f 1 be the limit of a sequence of points (1) » * * 1* 2,*** Briefly Assert that (3) p 6 . To prove (3 ) let (4) Qn~k be any open oriented (n-k)-cube of side-length s and suppose (5) s (6) Tt*p € Qn“k . Statement (2) implies (see 1.3.8 ) that ^Pm ^ p as m ■—> oo , which in turn implies by (6) the existence of an index m such that o (7) Qn"”k for m > m . m o Now (7), (5), (l) require (see II.9) that (8) I^_k t Q n“k D K*(S 0 r ^ ) ] * (l-(l/i) )sn“k for m > : m o . But by hypothesis S is closed, and so applying 1.3.11 we note that (since by (2) we have TTpm"^ TTp) (?) u v u p C Q ^ n **(sr> r h p m) ] a Qn-k n k (5 n n- % p ) . Successive application of (9), 1.2.6 , (8) yields U° ) Ln_k[ Q n"k n A s C \ TT^p)] -2: LR_k lim sup t Qn*"k 0 K ( s O TT ’''K’Pm )'] 2l m in o - 48 - »■ 11m sup Ln^ic[QI>*^C O «*(S C\ tf ^wpB)] > " m > Wq > (l-(l/i)>B"lt . Thus (10) follows from (U),(!>),(6) and so (see II. 9) assertion (3) is valid. Finally, since (1) and (2) imply (3), the set Du is closed, and so the lemna is proved* II. 12. Lemma. If S is closed, then D is Immeasurable* Proof: By 11.10 and 11.11 it follows that D is the countable intersection of countable unions of closed sets, which (since closed sets are I^-measurable) is an L^-neasurable set (see 1.2.4 ), and so the lemma is proved* 11*13. Lemma. If S is closed, then S . D is L -measurable. n Proof * By 11.12 we have that D is Immeasurable, and thus since S is by hypothesis closed (hence Immeasurable) the difference S - D is In-measurable (see 1*2.4)• II.llu Lemma. If S is closed, then I^(S - D) » 0 * Proof: This follows at once from 11*13 and 11*8* 11*35. Lemna. If S is Immeasurable, then Ln(S - D) ■ 0 • Proof: Since by hypothesis S is I^-measurable there exists by 1*2* 9 a sequence of sets S^, i ■ 1,2,..., for which the following properties (1) - (3) hold* -U9- (1) Is dosed* (2) Sj. C S^+i C L S j (3) L (S - U S±) - 0. n i-l By (2) and II*S ve also hare (It) DiC. D | where D * D(HjS)j Di — D(®jSi) j i — ly2y*** * In view of (1) we apply II*lii to the pairs S^, and obtain (5) y s ± . D±) • 0 , i - 1,2,*** . How using (U) clearly hare the inclusions O O OO q O o o U s4 . d c vj s. - J U 4 C- u (s, - V.) , i-1 i-1 1 i-1 i-1 1 and so applying (5) we hare OO I°n (S-D) < L°a [(S - ^ Si) J ( S± • D) ] < < l J ( S . p , S±) ♦ L°n ( U S± - D) < 0 , - -n i«l 1 “ i-l 1 and therefore since L°n » 0 we have I^(S - D) - 0 • Thus the lena is proved* -50- 11,16. SECTIONAL DENSITY THEOREM. Let S be an Immeasurable subset of Rn. Then (see 11,1) L [S • n»(*,S)] * 0 a. » where the Intersection is taken over all possible proper subsequences if of ^1) •••> n | f Prooft Given an arbitrary proper subsequence w of ^ 1, n j it follows from 11*15 that (see II. U) (1) L (S - D(t»,S)] • 0 . n But clearly we have the inclusion (2) S - C\ D(*,S) C . U [S - D(«,S)] , « * Now, since the set of proper subsequences if is finite, relations (1) and (2) yield L®tS - flD(*,S)) < 22.X*®CS-D(ir,8)J - 0 , n w . . * a and so since l£ » 0 we have y s - Q d (*,S)] - 0 ♦ Thus the theorem is proved* -51- III. THE ACCESSIBLE SET 111.1. Defirtition. Let S be any subset of Rn , and let p, x be any two points of S. Put j — (^> •••> x*^» P*^ > •••» P ) t j = •••> n — 1. Then x is said to be (p,S)-accessible if i j c S for j — 1,.. ,,n. For a given point p £ S the set of all (p,S)-accessible points is termed the accessible set of p, and is denoted by A^p,Sj. Note that A£p,S^ is a subset of S, and is defined only for points pt;S. 111.2. Lemma. If S^ t— S2 ‘ Rn , then p e: implies AfpjS^ CZ A[p,S2] . Proof: Let p t, S^ be given, and take any point x eA | p » sl]- Then (in the notation of III.l) Si for j = 1, ..., n , v and hence since S^ Sg we have t S2 for 3 1 j .. • f n . Accordingly (see III.l), x t= A|P>S2] , and so the lemma is proved. - 52- 111.3. Lemma. If S is a subset of the real-line R^, then pb S implies Ajp,sJ — S . Proof: By hypothesis n — 1 and hence (in the notation of III.l) we have 5-l - £ n •=. x t and so if p b S it is clear that x t A[p,S] if and only if xt: 5, and so the lemma is proved. 111.4. Lemma. Let n ^ 2 . Then given pfc S C R n the following formula holds (see 1.3.4 for notation): a[p,s]= (a[it£p, s(pn)] x Rl)f\ s . Proof: Let (1) P 6 S C R n . Note then that TT^p £ S(pn) , and so the accessible set is well-defined. Assert first that (2) A[P,S] C (AHtt*P, S(pn)] X R1) O S. To prove (2) take any point (3) x£;A^p,Sj| . Then (see III.l) (4) x (; S , and (5) (x^,...,x^ , p^ , ..., pn) CrS , j — 1, ..., n - 1. -53- From (5) it follows that (6) (x1, p''*1 , ....p11"1 ) ^ S(pn) , j = 1, n - 2 , (7) (x1 x”-1 ) 6 s(pn) . In view of (6), (7) we have (see III.l) (x1, xn-l) £ a[ tt*p, S(pn)] , and so clearly (8) x = (x^, ..., xn) fcr a [tt£ , S(Pn)] X R'1’ . Thus (3) implies (4) and (8), and so assertion (2) is true. Assert now that (9) S(pn)] X R1) 0 S C A[p,s] . To prove (9) take any point (10) x £ S such that (11) x £ a XTT^p , S(pn)] X R1. From (ll) it follows that (1 2 ) (x1, x"-1) <= *[n*p, s(pn)] - 54- and so (see III.l) (13) (x1 x"-1) e S(p") , (14) (x1 x t PJ+1, pn-1) 6 S(p") , .] = 1 Now (13), (14) respectively imply (15) (x1, ..., X11"’1, pn ) € S , (16) (x1, x3, pJ+1, P11”1, pn) e s , j -1) • • • , n -* 2. Thus (15),(16) yield •«• j n - 1 , and so in view of (10) we have (19) x 6 A[p,S] . Accordingly (10), (ll) imply (19), and so assertion (9) is true and the proof is complete. III.5. Lemma. Let S be a Borel (closed) (open) subset of Rn . Then given any point p £ S the accessible set A^p,S^J is a Borel (closed) (onen) subset of Rn . Proof: If n = 1 the lemma is a oonsequence of the identity (see III.3) k\p,S] S , since by hypothesis S is a Borel set. We proceed by induction on n. - 55- Let n 2 2 be a given positive integer, and let (1) S be a Borel subset of Rn Take any point p € S. Assume that (2) If S* is a Borel subset of Rn”^ , then p* €: S implies p ,S I is a Borel subset of R . We observe from (l) and 1 .3.4 that the section (3) S(pn) is a Borel subset of Rn‘“^'. From (2) and (3) it follows that (since TT^p € S(pn) ) (4) a £ tT*p , S(pn)^j is a Borel subset of Rn“\ But III.4 yields the identity (5) A[p,s] ^ U l > * p , S(pn)] A R1 ) H S. As a result of (5 ), (4), 1.3.18 * 1.2. 4 (or 1.1.8 in the closed and open case) it is clear that A[p,s] is a Borel subset of Rn , and so for the case of a Borel set the lemma is proved. Substitu tion of the words ''closed" and "open" for "Borel" in the above proof completes the proof of the lemma. III.6 . Lemma. Let S be an Ln-measurable subset of Rn . Let 1 n p =r(p ,...,p ) be a point of S satisfying the property that for n ^ 2 the section S(p^,...,pn) is Lj^-measurable for ks2,...,n. - 56- Then A£p,S^| is Immeasurable. Proof: If n = 1 the lemma is a consequence of the identity (see III.3) A^p,S] “ S , since by hypothesis S is Immeasurable. We proceed by induction on n . Let n ? 2 be any given positive integer. Assume the valid ity of the lemma when n - 1 replaces n . Let (1) S be a Ln-nieasurable subset of Rn. Take any point p =: (p^-, ..., pn) S satisfying the property that (2) S(pk,...,pn) is Lk_-j-measurable for k = 2, n . Note that (see I.3.A ) we have the identity (3) (S(pn)) (pk,...,pn_1) = S(pk,...,pn) for ks2,...,n - 1 . From (2) it follows that (put k * n ) (4) S(pn) is a Ln_ m measurable subset of Rn ” ^ , and from (2), (3) we have (5) (S(pn) ) (p^,.. .,pn"’^') is Lj^-measurable for k = 2,...,n-l . Now since TT*P = (p^>• • •>pn**^) *= S(pn) it follows from (A),(5)> and the inductive hypothesis that (6) A^IT*p,S(pn)^ is Ln-measurable. - 57- But by III.4 we have the identity (7 ) a[p ,s] = U [ n £ p , s ( Pn)] a r1) f\ s . As a consequence of (7),(6), 1.3.17 , (l), and 1.2.4 it follows that A[p,s] is Ln-measurable. Accordingly, the lemma is proved. III.7. Corollary. Let S be an Ln-raeasurable subset of Rn . Then for almost every point p € S the accessible set a[p,s] is Ln-measurable. Proof: Since by hypothesis S is Ln-measurable it follows from 1.3.14 that almost every point p fz S has the property that S(pk,...,pn) is Lj^-measurable for k — 2,...,n. So application of III.6 completes the proof. Ill.8* Lemma. Let S be an L^-measurable subset of Rn, and let p “ (p , •••>p ) be a point satisfying the following conditions: (i) p e s , (ii) p 6 A * S (see 1.4.2 ) . (iii) For n — 2 and k = 2,...,n (see 1.3.4), (p1, . . . ^ - 1) 6 A ( S(pk,...,pn) ) . (iv) For n 2 2 and k — 2,.. .,n the section S(pk,... ,pn) is Lk_i-measurable. Then p e A( a [p ,s ] ) Proof: If n s 1 the lemma follows from (li) and the identity (see III.3) A^p,S"j = S . We proceed by induction, on n. Let n S 2 be a given positive integer. Assume the validity of the theorem when n - 1 replaces n . Let (1) S be an Ln~measurable subset of Rn . Take any point p (p^,...,pn) such that the following conditions (2) - (5) hold: (2) p £, S, (3) p € S, (4) < A ( S(pk,...,pn) ) for k r 2,...,n , (5) S(pk , ...,pn) is Lk_-j-measurable for k=2,...,n . We now consider the projection TTnP =( p \ ...,pn”"^) and the sec tion S(pn) , and note first that by (2) we have (6) Tr£p € S(pn) . From 1.3.4 we recall the identity (7) ( S(pn) ) (pk,...,pn) = S ( p k,...,pn) for k s 2,..., n-1. Then using (7) we obtain from (4),(5) the following statements (8), (9), (10), (11) . (8) n'p e A ( s(p") ) , -59- (9) (p1,...,pk-1) £ A [ ( S(p") ) (p*,...^"-1)] for k - 2,..., n - 1. (10) S(pn) is Ln_^-measurable , (11) ( S(pn) ) (p^,...,pn is L^_-^-measurable for k = 2, • • •, n "™ 1 • Now noting (6), (8) - (ll), we have by the inductive hypothesis that (12) TT^P <= A ( ALTT*p, S(pn)] ) . Observe that (10), (ll), and III. 6 imply that A^H^p, S(pn)^ is Ln_-^-measurable, and so from (12) and 1.4.8 it follows that (13) p e A ( A |[TT*P|S(pn)3 x r1 ) Noting (3) and (13) we apply 1.4.5 to obtain p 6 A [ (Alirjp,s(pn)] ^ r1 ) O s 3 , and so by the identity of III.4 we obtain p 6: / A ( A £p,S3 ) . Accordingly, the lemma is proved. III.9. Corollary. Let S be a Borel subset of Rn, and let p = (p^-,...,pn) be a point satisfying the following conditions: (i) p 6 S, (ii) p & A S . - 60- (iii) For n "2 2 and k r 2, ...,n , A ( s(pk,...,Pn) ) Then P fc: A (A\'p,S'] ) . Proof: This follows at once from III.9 since (see 1.3.15 ) condi tion (iv) of III.8 is always satisfied when S is a Borel set. III.10 ACCESSIBILITY THEOREM. Let S be an Ln-measurable subset of Rn. Then p Z^(A[p,S] ) for almost every point p of S . Proof: Note first that condition (iii) of III.8. is satisfied if and only if (see 1.3.4 and II.1 ) n P & O D(lT<,S) . J=2 Now let S* denote the set of points p for which condition III.8(iv) is satisfied. Then if E denotes the set of points p for which conditions (i) - (iv) of III.8 are satisfied simultaneously, we have Er S n s"0 ( As)fl[ r \ D (n*j,S) ] . 0*2 Since clearly S - E C (S - S*) U (S - A S) U U S - ^ D( ,,S) 1 J 5 2 J - 61- it follows that (1) L^S-E) £ Ln(S-S*) + 1^(3- Z^S) + + h S . s - r \ d ( > v s ) i • J--2 But by 1.3.14 and 1.4.3 , respectively, we have (2) Ln(S-S*)= 0 , ^(S - Z \ S) - 0 . Also, as a result of the Sectional Density Theorem (see II.16) we have (3) L n [ s - Y) D( j,S>3 - 1 ^ 0 - f|D(fl-,S)] - 0 j=2 IT Accordingly, in view of (l), (2), (3) we have Ln (S-E) — 0, so that since 1^ is non-negative (4) Lyj (S - E) - 0. Now by the definition of E it follows from III.8 that every point of E has the property P ( A[p,Sl ). So in view of (4) the proof is complete. - 62- 17. THE RAUEMACHEEUSTEPANOFF THEOREM IV.1. Definition. Let f be a real-valued function defined on a subset S of R& and let p £ S be a point of accumulation of S. Then f is said to hare an S-relative total differential at p, if there exist (finite) numbers A^, ..., A^ such that f(p) - f(x) - A 1(p^-xJ) lim i-1 J x<= S, x - ^ p U P - x i\ exists and is equal to zero* (Note that if the exists, we have A j - f*(p) .) 3 J IV.2. Corollary. Under the hypotheses of 1 7 ,1 , f has an S-relative total differential at p t S if and only if there exist (finite) numbers A_,...,A such that for any assigned <= » 0 there exists a S » £> ( ^ ) x n > 0 such that n I f(p) - f(x) - S A (p^-x3)| < 6 Up - xU j-1 J if x t S , ll P - x U < S • 17*3* Lemma. Let f be a real-valued function defined and continuous on a bounded open subset S of R1*. If f has S-relative j-partial derivatives, j -1, ..., n , at almost every point of S (see 1*5*10 ) then for any assigned nurfcers Q > 0, > 0 there exist a set E ■ E( t , ^ ) and a number S » 5 ( fc , ^) > 0 for which the following statements hold: ( i ) E C Sj (ii) E is closed; (iii) L ( S - E ) < £ s n (iv) The S-relative j-partial derivatives fj, J ■ 1, n , exist at each point of E j (v) If (a) p £ E , (b) x ^ A[p,E] (see III.l ), andilp - xU <& , then lf(p ) - f(x ) - 3 fl(p ) (p^-acJ)! < v U p * xU. j»l 3 & Proof: By hypothesis there exists a subset S* of S such that (1) Ln( S - S*) - 0 (2) The S-relative j-partial derivatives, j • 1, n, exist at * each point of S • Now f is by hypothesis continuous on S and by implication (see 1*5. 11) the S-relative j-partial derivatives are each L^-measurable.. on S* . ♦ Therefore, slnce(in vianrof (1) and the openness of S) S is a bounded it L -measurable set. we can select (see 1.5.8 ) a set E for which the n following statements (3) - (6) hold: (3) E* Cl S* j * (ll) E is closed; (5>) L (S* - E*) < 6/2 } a (6) f, fj , j ■ 1, ..., n , are continuous on E* ( hence uniformly continuous, since E* is bounded and closed) • -6U- From (6) and 1.5.4 , there exists a number ^ % ( y ) > 0 such that (7) IfjC*^) - I ^ 2 n , j = 1, ..., n, if ^ , x2 £ E* and ll X]_ - X2 |1 . Now for each positive integer k denote by E ^ the set of points p for which the following statements (8), (9) hold. (8) p £ E*, (9) For given j ^ n , \f(p)- f(x) - f'.(p) (p^-xj)l^ (^/2n) | p^-x-31 0 if (a) x <= S , (b) fi^p" T*"^x (see 1.3.4), and (c) \ p^ - x^ \ -< l/k . First we note that clearly (10) ^ * k ""1, 2, ... . We next assert that (11) E* is closed , k - 1,2,... . k To prove (11) let pm € E^ be any convergent sequence of points, say Pm — > pD as m — . Then since E is closed and E^ d E we have pQ ^ E . Now let j — n be given and take any point x £ S with ^ P 0*"^"^x anc* | Po^ “ x*^| i ^ j i \ p0>) ~ x \ ^ we have 1P^ “ x I < l/k for a large enough, say f t m "z. m 1^ • Accordingly, since Pm ^ E^ » we have (12) |f(pj - f(S ) - f ’(p„) (pi - ^) l < ( IS/2») I pi - *i i for m > max (a^ , a £ ) • Now, since by hypothesis f is continuous on S and (by (6) ) each f' is continuous on E* , passage to the V limit in (12) yields lf(pc ) - f(x) - f^(p) (p^ - x?)\ < (IS/2n)Ip^ - . Thus (see (9) ) p ^ t , and so since pm was an arbitrary sequence assertion (U) is true* Now, since E*C. S* it follows; from (2),(9), and 1.5* 10 that # CXJ t (13) E • U e J - livn . k»l K Relations (10), (11), (13) imply (see 1*2*6 ) that there exists a positive integer k ■ k (fc) such that o o (lb) - E ^ ) < e/2 . Define (15) S - a i n ( S*, W , E - B*^ * We assert that S , E as defined in (15), satisfy the conditions (i) - (iv) of the lemma* Note first that conditions (i),(ii),(iv) are obviously satisfied in view of (10),(11),(2),(3)* Moreover, (iii) holds since b y (1), (5), (lb) we have LjS-Ej^) < Ln (S-S*) ♦ I^(S*.J£*) ♦ ^ ( E * - ^ ) < £ * -66- We proceed to verify (v)* Take any point p 6 E and (16) x € A[p,E) such that (17) ll p - x U < S . For brevity set p « S0 , x « , and (x1 , * . . ^ , p^+1,...,pn) - 3 ■ •••> n - 1 * Then by (16) and 111*1 we have (18) E , 3 ■ 0, l,***,n * Now, upon addition of suitable quantities involving £ we obtain J by the triangle inequality the following inequality (19) I f(p) - f(x) - Z f*(p) (p3-*J) \ < j-1 ^ ♦ 2 D | f ( f ) - f*(p) I . \ p^-r3 | • j-2 3 j-1 i We estimate the two summands on right side of inequality (19) as follows* Since by (18) we have 5 K and clearly * a js ■ j-1 J 5 w4 for j ■ l,..«,n , it follows from (9)> (l5) and (17) J j-1 that -67- (20) 2 2 |f(g ) - f(5 ) - f'(§ )(pj-xj)|^(l72n)2I|pj-xj l & j=l j-1 ^ j j“l j-1 ± ( y/2) ip-zt . Again since «^j_l ^ E and N^j-1 “ p W * lx^ ~ P^l for js2>***>n> it follows from (7),(15), (17) that (21) lfl(5 ) - f'(p)l. lPJ-«J\ffr/2n)2|pJ-xJl £ j-2 J J J j-2 — ( if/2) 11 p - x || . Accordingly, applying (20), (21) we obtain from (19) the inequality (22) |f(p) - f(x) - f (p) ( p j - x ' M ^ v||p - x \l j=l j Q Since thus (2l) follows from (16), (17) condition (v) is satis fied. Therefore the lemma is proved. IV.4. Lemma. Let f be a real-valued function defined on a bounded open subset S of Rn . If f is Lipschitzian on S, then f has an S- relative total differential at almost every point of S . Proof: Assign any £>0. For each positive integer m select numbers € m > 0 » ^ m>0 such that u) 6m-> o , irm —» o, ZL em < e . Since by hypothesis f is Lipschitzian on S, it is a fortieri contin uous on S (see 1.5*13), and (see 1.5.16) the S-relative j-partial derivatives exist at almost every point of S, j — l,...,n. Hence ( S being a bounded open set), the hypotheses of IV.3 are satisfied. — 68 — Accordingly, denote by E , the set £ and the number > o HI S J* S which correspond to <= • in the sense of IV* 3 • Put o o (2) B - C\ E" * n-1 m Then E, as the intersection of closed sets (see IV.3(ii) ), is closed and hence (3) E is L -measurable. n Furthermore, since clearly oo S - E c. u (S-E) ■4. * and (see IV.3(iii) ) , there follows in view of (1) the inequalities o q o o do ys-E) < Z 3 ys-E*) < 23 <= M < <= . B*1 1*1 Also (see IV.3(iv) ) the S-relative j-partial derivatives ^j(p) t j ■ l,...,n exist at each point p of E . In fact, since A[p,E] C _ A[p,En] for each p c: E(see 111*2 ) we have the inequality (see IV*3(v) ) (5) I f(p) - f(ac) - Z 3 f .(p) (p^-oc5) 1 < ll p - x W , j«l J * if p t E , x «s A[p,E] , and U p - x d < Since by hypothesis f is Lipschitsian on S, there exists a (finite) constant 1 A > 0 such that (6) \i’(p1) - *(P2)\ < M ^ Px-P2tt > ** ^ S * Take any point p £ E such that (7) p A a M . Then hy I,iu14 (wherein we put » / ^ n M and S • A[p,E] ) there exists for each m a number S * • S>* (p, Y m) 811011 that given any point qfc S with ll p - q U < S * there corresponds a point qn, G A[p,E] for which we have the inequality (8) ll - qll < ( ^ / n M ) II p-qll. Define ^ c sM, s * ) , and take any point (9) q 6 3 with Up-qU < & * * • Ve assert that under condition (9) we have the inequality (10) \f(p) - f(q) - z l x f (p) (pJ-q^H < lip - qli • j-1 ^ ** To prove (10) select for each n a point q^ *= A[p,E) such that Inequality (8) holds* Upon Intruding terms involving we obtain by the triangle inequality the followingr (11) I f(p) - f(q) - ^(p)(p^-q^)l < I *( -70- ♦ U(p) - f(qj f*(p) (pj - qj) I ♦ j-1 3 ♦ (p^-q£) - fj(p) (p^-q^)i • We now estimate the three stuanands on the right hand side of inequality (11) • For the first term we apply (6), (8), (9) and thus obtain the inequalities (12) [ fCq^) - f(q) 1 < M iiq^ -qU < < M ( Y /nM ) 11 P - q a < n < U p - ql‘ • For the second tern note first that by (8), (9) (assuming without loss that u < 1) we have the inequalities„ Up - \ U < Up - qU * ^ - q u < 2 Up • qlk < < ■ ~ Accordingly, since by choice q^ ct A[p,E], we may apply (5) and thus obtain the inequalities (13) U(p) - « < n - 2 fj(p) a p - < i 2 if* U p - q u • To estimate the third term we first observe that (6) requires that -71- \ fj(p)\ — M, j=Sl,...,n . Therefore, using (8) we obtain (14) 2 2 if ,(p)(p - % ) - fj(p)(p'5--<^)I — j-1 m M | 1 ~ j-1 m tr n M II q - a || t£z. m ^ UmY II p - q\l. Finally, applying estimates (12), (13), (14) to inequality (ll) we obtain the asserted inequality (10). Accordingly, (9) implies (10) and so since m -**> 0 the function f has on S-relative total differential at p (selected as in (7) ) . But since by the Accessibility Theorem (see III.10) (7) holds for almost every p 6 E it follows that (15) f has an S-relative total differential at almost every point of E . Thus, in view of (3), (4 ), and (15), application of 1.2.12 com pletes the proof. IV.5. Lipschitzian Extension Theorem (due to MacShane [5b] ) : Let f be a real-valued function defined on an arbitrary subset S of Rn. Assume that f is Lipschitzian on S. Then there exists a real-valued function f with the following properties: -72- , * n (i) f is defined on the whole space H 9 (ii) f * is Lips chit sian on H11 , (iii) f * (p) ■ f(p) for p € S • Proof: By hypothesis there exists a (finite) constant M > 0 such that (1) If(P.) - f (p J I < K lip. • Pjt 1 2 1 2 n for every pair i l , p <= S. Define for x (2) f*(x) « l.u*b. (f(p) - Mllx-pli) • p€ s We proceed to verify that f satisfies the requirements (i),(ii),(iii), n Choose a point p^ <=. S* Let x be any point in R ♦ For p c S ve have then (see (1) ) f(p) - MU x - pit » f(p0 ) ♦ (f(p) - f(pQ) ) • M U x • pU < < f(p ) + M U p0 - p U-MUx-pil# — o Since H > 0 and U P - Ptt - tV at - p IV < VI P0 - x it by the triangle inequality, it follows that (3) f(p) - M 11 x - p It < f(p0 ) ♦ M H pc - x U for p 6: S * From (2) and (3) we conclude that (It) f*(x) < f(po) + M U p ^ . x U . -73- On the other hand, (2) yields (5) f*(x) > f(p0 ) - M |l x - PQll • From (U) and (5) we see that f*(x) has a finite value at each point of Rn , and (i) follows* Next we verify (ii) by showing that (6) | f (x^ - f*(Xg) \ xi n for every pair x^, x^ ^ R * Select any two points x^, Xg6 R * Let the notation be so chosen that (7) f*(x^) - f# (x1 ) > 0 Assign 6 > 0. In view of the defining formula (2), there exists a point p# S such that (8) f*(Xg) < f(p#) - M U Xj - p* tt ♦ e • On the other hand* (2) yields (9) f * ^ ) » f(p*) - M ttx2 - pj\ . From (7)» (8)* (9) we obtain the inequalities 0 < f*(Xg) - **(*3) < Observe that W x2 - p# \l • - p# U < U x g - XjW by the triangle inequality* Hence* since M > 0 , -7U- (10) 0 <-f*(Xj) - < <& ♦ ¥ U *2 - . As (=. was arbitrary, clearly (10) implies (6), and thus (11) is proved* Consider now any point p <~ S* Replacing x by p in (U) o ° and (5) we obtain the inequalities f (PQ ) < f*(P0 ) 5 f (P0 ) and (iii) follows* 17*6. Theorem* Let f be a real-valued function defined on a n bounded Immeasurable subset S of 9 * If f is lipschitzian on S , then f has an S-relative total differential at almost every point of S * Proof: Let Q be an open n-cube containing S, and in accordance in . j«. * 17*5 let f denote the extension of f to Q such that f is Lipschitzian on Q* Then by IV,U f has a Q-relative total differential at almost every point of Q and hence at almost every * point of S* In fact, if f has a Q-relative total differential at a point p <= S, then, unless p is an isolated point of S, it is clear (see 17*1) that f has an S-relative total differential at p* But (see 1*1*10 ) the set of isolated points of S is countable, a * fortiori of L -Measure *ero« Accordingly, f has an S-rel&tiro n * total-differential at almost every point of S, and hence since f agrees with f on S the theorem follows* 17*7* Definition* Let f be a real-valued function defined on a -75- n subset S of R * Then f is said to be of S-relative bounded linear distortion at a point p € S if there exist numbers M > 0 , S > 0 (depending upon p) such that lf(p) - f(x)\« M U P • x i if S, llp-xll < S • IV.8* Lemma. Let f be a reaL-valued L -measurable function defined n on a bounded L -measurable subset S of S11# If f is of S-relative n bounded linear distortion at almost every point of S, then for any assigned G > 0 there exist a set E ■ E(<= ) and two musters M » M( £ ) > 0 ,S"5"( (i) E C. S; (ii) E is closed; (iii) I^(S-E)< C f (iv) If (a) p G E, (b) x , then I f (p) - f (x) \ < M \\ p - x \\ ; (v) f is Lipschitzian on E « Proof: By hypothesis f is I^-measurable on a bounded Immeasurable set S, and is of bounded linear distortion at almost every point of S. Hence (see 1.5.9 ) we can select a set E for which the following statements (1) - (5) hold: (1) E*ci s , (2) E* is closed, (3) I^(E-E*) <€/2 , -76- (li) f is continuous on E , (5) f is of S-relative bounded linear distortion at eachpoint of E • * For each positive integer k we denote by E^ the set of points p for which the following statements (6) - (8) are true* (6) p <= E* , (7) I f(p) I < k , (8) lf(p) - f(x) 1 < k \Vp - x l\ if i C: S and ftp«xtf At once we note that clearly (9) Ej[ 1!* $ k ■ 1|2)(«« • We next assert that (10) Bu is closed, k * 1,2,••• To see this take any convergent sequence of points p 6 1 , say n » p — * p_ * Then p 6r £* , since E* is closed and E? E** a ' o o k Also by (7) we have the inequality \ f(pa) I < k and so in view of (li) we have 1 f(pQ) \ < k* Now take any point x <=r S such that U pQ • x < 1A • Then by the continuity of the distanee- function we have U p^ • x < l A for a large enough, say a > and hence by (8) I *(Pa) • *(*) I < kllp^ - xtt for ■ > * 0 • By the supposed continuity of f (and of course the distance-function), passage to the limit yields -77- |f(p ) - f(x)| < k 11 p - Xll . o ° Accordingly (see (6)-(8) ) , p G E. and so since p was an arbitrary o * n convergent sequence assertion (10) is true* Ve now assert that CM) (11) E* - U E* * k-1 To prove (11) take any point pQ<= E * Then by (5) there exist numbers M > 0 , % > 0 such that (12) lf(p0 ) - *(x) I < K || p^ • x U if x £ S , |lp0 • x|V < 3 * Select a positive integer k such that (13) |f(P0)l < k , M Then by (12) and (13) lf(po ) - f(x) | < M V\P0 • < k U P0 - xll ifftp0- x 4 < ^ »a fortiori if ll p Q • x ll < 1/k • * * * Thus p0 Ejg , and so sincep was arbitrary EC- U E ^ * * * / k"l Accordingly, since each E^ E , assertion (11) holds* Now, in -view of (9), (10), (11), application of 1*2*6 yields a positive integer kQ - k^fe ) such that (1U) y E * - Ej^ ) < 6/2 * Let us put (15) E - E ^ , M - 2kg , % - 1 A 0 . -78- We assert that E, M, 5 as defined in OS) satisfy the conditions (i) - (v) of the lemma* Note first that conditions (i),(ii), (iv) are obviously satisfied in view of (9)>(10), (8)* Also (iii) holds since by (3) and (lU) we have y S - E ^ ) < LjS-ti*) ♦ y E #-E^ ) < €■ • We proceed to verify (v)* Take any two points p,q fc E* If U p - q U < S' then (since S » lAo) property (8) yields I f(p) - f(q) t < U p - q ll < 2kJ H p - q \l • Suppose now that U p - qll > 5" • Then by the triangle inequality and property (8), we have |f(p) - f(q) I < |f(p)l ♦ lf(q) I < 2ko - 2kJ (lA0) £ 2k* lip - q U . Accordingly, (seel.S.**) (v) is verified, and so the lemma is proved* 17*9* Theorem* Let f be a real-valued I^-measurable function defined on a bounded L -measurable subset S of R11* If f is of S-relative n bounded linear distortion at almost every point of S, then f has an S-relative total differential at almost every point of S* Proof: Let any £ > 0 be assigned* Then in accordance with IV*6 let E ■ E( ^-) be a set and let <=) >0, M > 0 be two numbers for which the following statements (1) - (1;) hold: (1) E is a closed subset of S , -79- (2) y S - E ) < (3) f is Lipschitzian on E j (U) If (a) p <=■ E , (b) x «= S, and (c) Up - xll < 5* , than lf(p) - f (x) i < M Up - xll . In view of condition (3) and IV*6, we have that (£) f has an E-relative total differential at almost every point of E* We assert that f has an S-relative total differential at almost every point of E. To see this take any point p e E satisfying the two conditions (6) p e /S. e , (7) f has an E-relative total differential at p * In view of (7) and (U) it is clear that the E-relative j-partial t derivatives f^(p) exist and (8) 1 * j (p ) I < M , j - 1, n * Let 0 be an arbitrarily assigned number* In accordance with t t (6) and I«U*14 let <5 - ^ (p, ^ ) > 0 be selected so that given any point q ^ S with U P - < S 1 there corresponds a point q^ ^ E such that (9) U q» - q U < ( g / n M ) U p - qj U ♦ Moreover, in accordance with IV*1 and (7), let - t>** ( ^ ) > 0 be selected so that if x €. E and | \ p - x U < S*1 then (10) |f(p)-f(x)- **(p )(p ^-*^) 1 < VttP-xU • 3-1 J * (where the fj(p) are E-relative) • Define (U) ^ win ( S,S > 5 " ) » and take any point (12) q £ S with Up- qU < We assert that under condition (12) we have (13) \f(p) - f(q) - ^ f^(p) (pd qJ) \ < U ^\\P • q » • To prove (35) select a point q^, E satisfying inequality (9) , and note that (assuming without loss that < 1 ) we have (1U) \\ P - q* \\ < \\ P - q u ♦ Uq* - qU < 2 U p - q U Upon intruding terms involving we obtain by the triangle inequality the following: (35) 1 f(p) - f(q) - 2 ^ fj(p) (p^-qJ) \ < \f(q*) - f(q)\ + - 81- ♦ \ f(p) - f(q^) - ^(p) (P^-qj[) I + ♦ 2 2 iU*( p ) (pJ-qj[) - f.(p) (p W ) 1 j-1 3 ^ 3 We now estimate the summands on the right hand side of inequality (15) . r- For the first term, since by (9), (11) we have H q* - q U < 3 < o> we obtain from (U) and (9) the inequalities (16) \f(q*) - f(q) I < Ml|^-q» < M ( tf/nM ) UP • qll < £ U P - q II • For the second term we observe from (1H) that since E it follows from (10) that (17) lf(p) - f(q*) - S fj(p) (p^-q^i < P • q»ti < < 2 ^ \\ p - q U , the latter Inequality following from (lib) « To estimate the third term we apply (8) and (9) to obtain (18) *'(p)(p*-q*) - *-(p)(p*-q3)l < j«2 3 * o < M^Zjlqd-q?l *nH llq-q*U < j«2 * < p - qU • Finally applying estimates (16), (17), (18) to Inequality (15) we obtain the asserted inequality (13) • Accordingly, (12) implies - 82- (13) and so since ^ was arbitrary f has an S-relative total differential at p (as in (6), (7) )• But since b y IV»6 and I«$.5 conditions (6) and (7) each (and hence both) hold for almost every point p 6 ! it follows that (19) f has an S-relative total differential at almost every point of E* Thus, in view of (1), (2), and (19), application of I»2«12 completes the proof* -83- IV.10. RADEMACHER-STEPANOFF THEOREM. Let f be a real-valued Im measurable function defined on a bounded L -measurable subset S n of R*1# Then a necessary and sufficient condition that f have an S-relative total differential at almost every point of S is that f be of S-relative bounded linear distortion at almost every point of S* Proof: The sufficiency is given in 17.9* For the necessity, assume that f has an S-relative total differential at a point p 6 S. Then (see IV#? ) there exists a numberS «S(l) >- o such that for every point x o S with il p - x U < S we have the equality lf(p) - f(x) - 2 fl(p) < IIP - x|l # 3-1 3 If we select any number M such that i M > 3 - !>♦•* n , we therefore have by the triangle inequality \ f (p) - f (x) | - Mn|| p - x\l < < if(p) - f(x) - 2 1 ^ ( p X p ^ x 5)! < 3-1 3 < II P - xll • Thus I f (p) - f(x) | < (ljrt>l) l\p-xl\ , if x6 S , llp-xU < $ , and so (see 27*7 ) f is of S-relative bounded linear distortion at p# Accordingly, since f has an S-relative total differential at almost every point p 6 S, the proof is complete# -8U- V. THE STEPANOFF THEOREM V*l. Definition. Let f be a real-valued I^-measurable function defined on a bounded Immeasurable subset S of R?. Then given any subset E^t 0 of S, a point E, and a number M > 0 we denote by E(p,M) the subset of E defined by the formula E(pjM) ■ ^ x \ x E , |f(p) - f(x)|< M ft p - xU^ . In particular, S(p,M) - { x \ x e S , 1 f(p) - f(x) I < M II p - x ij | • V.2. Lemma. Under the hypotheses of V.l , let there be given a subset E # 0 of S , a point p e E , and a number M > 0 * Then (see V.l) ve have the formula E(p,M) - E f\s(p,M) and hence the inclusion E(p,M) c- S(p,M). Proof: This follows at once from V.l. V.3. Lemma. Under the hypotheses of V.2 , if E is Immeasurable, then E(p,M) is Immeasurable. Proof: Let g(x) - 1 f(p) - f(x) \ - M U p - x ll • Then since f(x) is L -measurable, so also is g(x) (see 1.5.7 ). Therefore n since E is L -measurable and n -85- E(p,M) - [xl xCrE, g(x) < 0 \ , it follows from 1.5*6 that E(p,M) is I^-measurable, and so the lemma is proved* V*lt* Lemma* Under the hypotheses of V*2 , if E is closed and f is continuous on E, then E(p,M) is closed* Proof: Let g(x) » \ f(p) - f(x) 1 - M U p - xll * Then since by hypothesis f(x) is continuous on E, so also is g(x) (see 1*5* 5 } * Therefore, since by hypothesis E is closed and E(p,M) • [ x I x f c E , g(x) < 0 J it follows from 1.5*3 that E(p,M) closed, and so the lemma is proved* V.5. Definition* Given f as in V*1 and any subset E ^ (jf of S we shall say that f is of approximately E-relative bounded linear distortion (b*l*d*) at a point p c E if there exists a (finite) constant H > 0 such that (see Y*1 and I*lj*2) p 6 E (p,M) * V*6. Remark* If f is of approximately B-relative b.l.d. at a point p <= E , then p E * V*7* Remark* If p^r E O ZX E and f is of E-relative b*l*d* at p, then f is of approximately E-relative b.l.d* at p* V*8* Lemma* Given f as in V*l* and any subset E of S, if f is of approximately E-relative b*l*d* at a point p — 86— approximately S-relative b.l.d. at p • Proof: This is clear* since by V.2 we have the inclusion E(p,H) C - S(p,M) so that by I. A e (p ,H) <- Z^S(p,M). V.9. Theorem. Let f be a real-valued L -measurable function defined n on a bounded Immeasurable subset S of Rn. If f is of approximately S-relative b.l.d. at almost every point of S, then for any assigned > 0 there exists a set E ■ E(<~) for which the following statements hold: (i) E C S , (ii) E is closed* (iii) Ln (S-E) < e , (iv) f is Lipschitsian on E . Proof: Let C- > 0 be assigned. Then in accordance with 1.5. 9 since by hypothesis f is I^-measurable on a bounded Immeasurable set and f is of approximately S-relative b.l.d. at almost every at point of S* there exists a set E - E (<= ) such that the following conditions (1) « (5) hold. (1) e * c: S , , * * (2) E is closed* (3) %(S-E*) < C-A , (U) f is continuous on E -87- * (£) f is of approximately S-relative b.l.d. at each point of E • Let us recall (see I.U.9 ) that if p e Rn and r is any positive mufeer, then K(p,r) denotes the closed spherical neighborhood of p with radius r. Moreover, (see I.iul5 ), if p^, p2 are any two distinct n points in R , then the nuaber ^ 0( - M K(P1» * P2^» )Q k (P2» A V(P1> - pg.v) ♦ I^K (P2 , UPx-P 2 tt ) Is independent of the choice of pj, pj, and satisfies the inequalities (7) 0 Let p t S, and let M, r be two positive numbers. Observe that by . V.3 » I.U.10 , and 1.2.4 the set S(p,M) fl K(p,r) is Immeasurable. For each positive integer k, let ES£ be the set of those points p which satisfy the following conditions (8) - (10) * (8) p <= E* , (9) I f(p) I < k , (10) If r is any ntmfcer such that 0 < r < l/k , then ys(p,k) H K(p,r) ] > (1 - oc ) 1^ K(p,r) . Ve first assert that (11) E* - U E* k»l * -88- To prove (11), consider any point p0 €. E* • Since f is of approximately S-relative b.l.d. at pQ (see (f>) ), there exists (see V.£) a constant M > 0 such that pQ <=: ZhS(pQ ,M) • Accordingly, in view of (7) and I.iuU there exists a mnsber rQ > 0 such that (12) y s ( p 0 ,M) K(p0 ,r)] > (1 - e( ) y ( p Q ,r) if 0 < r < rQ . If k is a sufficiently large integer then (13) \ f(p0 ) \ < k , M < k , l A < r 0 * Let k be so selected that (13) holds, and let r be any number such that 0 < r < 1/k • Then clearly S(pQ ,k) ^ S(po , M) , and hence by (12) we have the inequalities y S(p0,k)n K(P0*r)J > y s ( p 0,M)n K(p0,r)l> (1-C() y ( P 0,r). Thus it is established (see (8),(9),(10) ) that pQ ^. , and so since pc was an arbitrary point of E* assertion (11) is true* We shall now verify that f is Lipschitzian on each E^« This fact will be proved if we show that for any two points p^,P2 *=. E^ we have the inequality ( H O Iftp^ - f(p2)l< 2k2 1\P!-P2\V • Since (lit) is obvious if p^ » p^ we can assume that p^^r pj> « If U Pi - P2 W > l A * (®i»ce (9) requires \ f(p)\ < k for p £ E£ ) -89- \ t ( p { ) - f(pj)l < \t(.P1 )\ * lf(p2 )|< 2k < 2k2(lA) < < 2k2 U Pi - P*ll » and so (11*) holds In this case* So we can assxune that (15) 0 < U P i - PgU < l A • Let us put (16) * S(p^#k) f Sg “ S(p2»k) t (17) ^ - K(p1, UPi-PgW ) , Kg - K(pg, JlPj. - PgU ) • In view of (10) and (6) we have then the relations (18) y S i O Kx ) > (1-cO 1 ^ , Ln (S2 0 K2 ) > (l-«) LnK2 , (19) U - ‘ ‘ “ • V i ♦ V a Then (in view of (16) - (19) ) there follows from I.Jj.16 the existence of a point (20) P ^ ^ ^ H S g such that (21) U V i - P * U < - P2\l , \\ P2 - fell < W Px - P2U • As a result of (20) and (16) we have (22) \ t ( p i ) - f(p#) I < k l| ?! • p* \t * -90- (25) I f(p2) - f(p#)l ^ k U p2 - p * H From the triangle inequality, (22), (23), and (21) it follows that (24) l±*(Pl )- r(p2 )l ± lf(P l ) - f(p^)l * \ f(p2 ) - f(p#) \ * 6: k( U p - p U + Up - p U ) ^ fr 2kUPl - p2ft ^ 2k2UPl - P2U . Thus (14) holds* Now define for each positive integer k (25) E£ — closure • Then (see 1*1*8) (26) (27) Ej d 1J^.1 Also, since *S£ o £* and E* is d o s e d we have that d E*, and so by (11) c o (28) E* = U 1C k-i ^ . Now in view of (26), (27), (28) application of 1*2.6 yields a positive integer k ^ k^G) such that (29) Ifc( E* * Ejco ) *<• *-/Z Define (SO) E - E ^ . Then at once we have by (5), (29) that (31) Ita( S - E ) £L Ll ( S - E * ) +. I*( E* - I J ) <. € • o -91- Finally we assert that f is Lipschitzian on E. For take any two points p, q £! E. Then by (30) there exist sequences pm , q ^ E ^ such that pm — *> p, q — ^ q. By (14) we have then - f<%>l - 21£ II Pm - %&. For m —T c>o there follows (in view of (4), 28) ) the inequal ity If(p) - f(q)\ - 2ko 1\ P - q\\ , and so f is indeed Lipschitzian on E. Accordingly, with (l),(28),(26), (31) the theorem is proved. V.10. Corollary. Under the hypotheses of V.9, if f is of approxim ately S-relative b.l.d. at almost every point of S, then for any assigned £ * 0 there exists a set E * E(C) for which the following statements hold: (i) EdS, (ii) E is closed, (iii) Ln(S-E)<£, (iv) f has an E-relative total differential at almost every point of E. Proof: This follows at once from V. 9 and IV.6. V.ll. Definition. Let f be a real-valued Ln-measurable function defined on a bounded Ln-measurable subset S of Rn . Then given any subset E ^ 0 of S, a point p €. E, a number M >■ 0, and a positive integer j 6 n we denote by E(p,M,j) the set of real numbers § which -92- satisfy the following conditions (see I*3#4 )t (i) There exists a point x 4 E such that tup • w x and * * , v 3 wA x - ^ * Briefly, ^ *A (E \ ' «A » i? ) • 3 3 3 3 (ii)Given x as in (i), ve have the inequality \f(p) - f(x)l < M I p 3 - I . V*12. Lemma* Under the hypotheses of V*ll, let there be assigned a subset E $ of S, a point p t E , a muriber M > 0 , and a positive integer j < n * Then (see 7*11 and 7*1 for notation) E(p,M,j) - **[E(p,M) r\ l i f ] * 3 i 3 In particular, * - -1 S(P,M,J) - *J[S(p,M)n Wj « p] Proof: This follows at once from 7*11 and 7*1, if ve note that 11 p - xll « IP3 - $ I if * a x * irA p , s* x ■ q* J 3 j 7*13* Lemma* Under the hypotheses of 7*12, we have the inclusion E(p,M,j)C: S(p,M,3) . Proof: This follows at once from 7*12 and 7*2, since (see 1*3*6 ) * « a is monotone* j 7*llu Lemma* Under the hypotheses of 7*12, assume that E is closed and f is continuous on E* Then E(p,M,j) is closed* Proof: This follows at once from 7*U , 7*12, and 1*3*15 * -93- V*l5. Lemma. Under the hypotheses of V.12 assume that E is closed and f is continuous on E* Let there be given a sequence of points Pb £ E such that Pm — > p as m — ? <=>° * Then (see V.ll) lam sup E(p ,M, j)<£_ E(p,M,j) * m ■ Proof: Let (1) <% «c lim sup E(p ,M,j) • * m m Then (see 1*1*1 ) there exists a subsequence of * such that (2) fc E(p^,M, j) for each m^* Hence (see V*ll(i) ) we have (3) S ^ *t(E ft "I1 t o r each % > and so by I*3*H (since by hypothesis E is closed, and p ^ -> p implies wja ) it follows that J ^ (k ) % «= »*(E A n^1 n^p ) . j j ij Now, in accordance with (3) and (U), let x, x 6 E be defined by % the relations (5) y ■ V* Y “ § “d *?*■»' Y * » ’ * 3% " ? • Then by (2) and V*ll(ii) we have (6) | f(j^) - 1 < H - $ I • — 9U— But since p -- ^ p and hence by (5) (see 1.3.8 ) x — ^ x , jtim. 7 it follows from (6) and the continuity of f on E that (7) \f(p) - f(x)l d= M I - § 1 . Thus, (4) and (7) imply (see V.ll) that § € E(p>M,j), and so since § was arbitrary the lemma follows. V.16. Definition, Under the hypotheses of V.ll, given any subset E ^ of S and a positive integer j — n we shall say that f is j-partially of approximately E-relative bounded linear distortion (b.l.d.) at a point p E if there exists a (finite) constant M > 0 such that (see V.ll) pj £ Z*E(p,M,j) . If for j = 1.2,...,n the function f is j-partially of approximate ly E-relative b.l.d. at p, then we shall say that f is partially of approximately E-relative b.l at p . V.17. Lemma. Under the hypotheses of V.ll, let there be assigned a subset E ^ 0 of S and a positive integer j n. If f is j-partially of apnroximately E-relative b.l.d. at a point p € E, then f is j-partially of approximately S-relative b.l.d. at p. Proof: This follows at once from V.16, since by V.13 and 1.4.4 we have the inclusion AE(p,I,j) c As(p,M,j). -95- V,l8, Lemma* Under the hypotheses of V, 17, let p E such that p^ Z \ (E If f is j-partially of approximately S-relative b.l.d. at p, then f is ^-partially of approximately E*» relative b«l.d. at p* Proof j For brevity we write « instead of By hypothesis (1) p3 6 /Z* (E nm\ p ) . Since also by hypothesis f is J-partially of approximately S-relative b«l,d. at p (see V*l6 and V*12), there exists a constant M > 0 such that (2) pJ e ir*[S(p^f)n A p ] . Now, in view of (1) and (2) it follows from I«li*5 that (3) P3 6 A t - (E a nm\ p ) H n*[S(p,M)0 a"1* p ] ) But (see I.3*10 ) the set enclosed in the braces of (3) can be expressed as **[E f\ S(p,M) f\w*,,1ir p ] , so that (3) implies U ) p3^ A . «*[E f\ S(p,M)n n-3* p ] ♦ But by V*2 we have the formula E(p,M) - E 0s(p,M) and hence (it) implies -96- p^ £ [E(p,M) a w-1* p i * Accordingly (see V*l6 and 7,12), f is j-partially of approximately E-relative b.l,d. at p, and so the leans follows* 7*19, Lemma, Under the hypotheses of V,17, assume that E Is Im measurable, If f is j-partially of approximately S-relative b,l,d, at almost every point of S, then f is j-partially of approximately E-relative b,l,d, at almost every point of E* Prooft By hypothesis there exists a subset S of S such that (1) y s - S*) - 0 , (2) f is j-partially of approximately S-relative b,l,d, at each point of S* * Since by hypothesis E is Immeasurable, the Sectional Density Theorem (see 11*16 ), asserts the existence of a subset E* of E such that (3) L (E - E*) - 0 , n (U) p t E* implies p^ £ * 3 3 3 Now, since clearly E - E * f \ s * C (E - E*) U ( S - S*) , relations (1) and (3) yield L J E - B * n S*) < Lo (E - E*) ♦ y S - S*) - 0 , -9 7 - and so since 1^ is non-negative we have (5) L (E - K*f\ S*) « 0 . n But (U), (2)> and V,l8 imply that f is j-partially of approximately * E-relative b.l,d. at each point of E f | S « Accordingly, with (5) the lemma follows, V,20, Lemma, Under the hypotheses of V.17, assume that E is 1^- measurable. If f is Lipschit»ian on E, then f is j-partially of approximately of S-relative b,l,d* at almost every point of E, Proof: Since by hypothesis f is Lipschitsian on E, there exists a constant M > 0 such that (1) IfCpj) - f(p2)l < M U Pi - P2U , if Pi, P2 t E . Take any point p t E such that (2) p3 e /iaj(Ef\tt^p ) , B y (1) ve have that (see V,l) E - E(p,M) and so by V*12 ve have the identity (E p ) « E (p»M,j ) ♦ 3 3 Hence by (2) and 7,1 it follows that f is j-partially of approximately E-relative b.l.d. at p, and therefore (by 7,17) f is j-partially of approxiaately S-relative b,l,d, at p« -98- But by the Sectional Density Theorem (see 11*16 ) (since E is by hypothesis Immeasurable) relation (2) bolds for almost every point p ^ E , and so f is 3-parti ally of approximately S-relative b.l.d* at almost every point of E* Thus the lemma is proved* V*21* Theorem* Let f be a real-valued L -measurable function defined o* a b o ^ e d Immeasurable sebset S * k If , Is « a ^ ^ t e l , S-relative b*l*d* at almost every p-^nt of S, then f is j-partially of approximately S-relative b*l*;, *>st every point of S* Proof: This results at once by successive application of V*9, V*20, and 1*2* 12 * V,22« Definition* Let f be a real-valued L -measurable function n defined on a bounded L^-measurable subset S of R , let E f ft be a subset of Sy and let j < n be a given positive integer* Then for each positive inte^jr k let Ejj,. denote the set of points p for which the following statements hold: (i) P 6 E i (ii) lf(p) \ < k , (iii) Given any open interval I of length s < 1/k. with p3 <£ I there follows the inequality (see 7*11) L® I ir\E(p,k,j)] >2/3s. 7*23* Lemma* Under the hypotheses of V*22, for each positive integer k we have the inclusion ®lk.C E3,k*i * -99- Proof: Let k be held fixed, and take any point p £ Ejk • Then (see V*22(i),(ii) ) p interval I of length s < l/(k-t-l) • Then s < l/k , ao that since clearly E(p,k+l,;j) ~ZZ> E(p,k,J) we have (seeV.22(iii) ) Lf[ i r i E ( p , k U (j) ] ^ I O E(p,k,j) J > 2/3 s . Accordingly, p 6 Ej » and the lemma follows* V.24. Lemma* Under the hypotheses of V«22, assume that E is closed and f is continuous on E* Then for each positive integer k the set Ej^ (see 7*22) is closed* Proof: Let k be held fixed, and take any sequence of points (1) Pm € Ejjj. , m *1,2,..., with (2) Pm— y P as m — ^ c xD . Assert that (5) p £ Ejk . To prove (3) we first note that, sinoe by hypothesis E is dosed, statement (1),(2) implies that -100- (U) p E * Moreover, by (1) (see V*22(ii) ) we have \ f(PK ) 1 < k , so that by (2) and the assrawd continuity of f on E it follows that (5) U ( p ) l < k * i j Now let I be an assigned open interval of length s < k with p €~ I • Then by (2) we have dj- I for m large enough, say m > n^. Thus from (1) and V*22(iii) it follows that (6) L ^ i n E(pm ,k,j)] >2/3s for m > uy, * Bub by V*l5, since E is assumed closed and f continuous, lim sup I n E(p ,k»3)C lflE(p,l,j) , ■ w , Hence, since each E(pR,k,j) is closed by V*1L, a fortiori Im measurable, it follows from 1*2*6 that (7) Ij[lAE(p|kii)] > 11m sup ♦ nwflo From (6) and (7) we obtain (8) Litin*(P»k*J)] >2/3 s . Accordingly, (1) and (2) imply (U),(5), (8) and so (see V*22) assertion (3) is true, and the lemma follows# V*25* Lemma* Under the hypotheses of V*22, assume that f is j- partially of approximately E-relative b*l*d. at almost every point of E* Then (see Y*22) -101- Proofs By hypothesis there exists a subset E of E such that (1) (2) f is ^-partially of approxiwately E-relative b.l.d. at •Jj. each point of E • Assert that 0 3 (3) E ^ O Ejjj. k - 1 J To prove (3) let pQ C E be given* Then by (2) and V,l6 there exists a (finite) constant M > 0 such that p3 ^ ZJw E(p,H,J) , o and hence (see I*U#7 ) there exists a nunfeer s0 > 0 such that for any open interval I of length s with pjj 6: I we have (10L ^ I 0 E(p0 ,M,j)) > 2/3 s if 0 < s < s0 . If k is a sufficiently large Integer, then (S) I *(PC ) l Let k be so selected that (5) holds, and let I be any open interval of length s such that 0 < s < 1/k* Then by (it), since dearly E(p,k,j) ~^E(p,M,j), we have the inequalities I1 [inE(p0,k,j)] > I1tlAE(p0,M,j)3 > 2/3 S. Thus it is established (see V*22(i)-(iii) ) that pQ € and so, since pQ was an arbitrary point of E* , assertion (3) holds* Now (3) and (1) yield * t ( E - U E 1k ) < l A E - E ) - 0 , n ic-1 J and hence, since 1^ is non-negative, it follows that o O L (E - (J E, ) - 0 , n k»l and the lemma follows* 7*26* Lemma* Under the hypotheses of V*22, for each positive integer k the inequality | f (p) - f(q) \ < 2k2 1 p^ - q^ \ holds whenever p, q 6 E and « p • ® q • & 3 3 Proof: Let k be held fixed, and take any two points (1) p, q <£ E, 3k such that (2) P - * q • 3 5 Assert that (3) U(p) - f(q)l < 2k2 lp^ • q^ \ * Since (3) is obvious if p^ » q^ (since then (2) implies p ■ q) we - 103- may assume that • First suppose (k ) \pJ-qJ\ > l A . Then, b y the triangle inequality, (1), V,22(ii), and (U), we hare the inequalities U ( p ) - f(q) \ < [f(p)\ ♦ \ f(q)l < 2k - 2k2(lA) < < 2k2 \ p3 - qJ l* Nov suppose (6) 0 < I p^ - qJ I < lA • Let Ij, I 2 denote the open intervals of length s * 2 \ p^ - q^ \ and with centers at ^^respectively. Then by (1), (6), and 7,22(lii) we have the inequalities ^ Clx fV E(p,k,3)3 > 2/3 s, ^ [I2 nE(q,k,j)] > 2/3 s , Accordingly, we can apply I,l;.17 and thus obtain a number (7) $ E(p,k,j) f\ E(q,k,J ) such that (8) I p^ - 3 \ < \p^ - q^ \ , Vq3 - § \ < V P*5 - q^V • By (7), V.11, and (2) there is a point x £ E such that w* x ■ ^ , J x » it j q , and -IOI4- (9) lf(p) - f(x)l< k ip^ -§l , lf(q) - f(x)|< k \ q ^ - • By the triangle inequality, (9), and (8) we obtain the inequalities \f(p) - f(q)l< U(p) - f(x)\ + lf(q) - *(*)l< < k( lp5 -g\ * \q* - f\) < 2k^p3 - qj| < < 2k2 \ pJ - qj l • Thus assertion (3) is true, and so the lenana is proved* V#?7* Definition. Let f be a real-valued function defined on a subset S of Rn. Then f is termed uniformly partially Lipschitzian on S if there exists a (finite) constant M > 0 such that for 3 * 1,2,#.*,n the inequality \f(p) - f(q)l < M \p3 - q*5 \ holds whenever p, q 6 S and p ■ q « xv V*28. Lemma* Given f as in V.27, assume that S if L^-measurable. If f is uniformly partially Lipschitzian on S, then f is of approximately S-relative b*l*d. at almost every point of S* Proof: Since by hypothesis f is uniformly partial Lipschitzian on S, there exists a constant M > 0 such that for J ■ l,...,n we have the inequality (1) U(p) - f(q)) < M \ p^ • q^l, if p, q 6 S and Wj p - it j q . -ic5- Let p € S be given, and take any point (sae 111*1 ) (2) x e A(p,S] * Then upon setting S o ■ P* *„-* 5. ■ (l » •••) 3 j j P ) j i ■ Zj it follows from (2) that (3) S i € S for 3 Since clearly ■ n f for 3 - l,*..,n, there results from 3 j j-1 (3) and (1) the Inequalities (h) I fjp) - f(x) I < 2H1 *( £. •,) “ < 3-1 3 n j . < Vp3 - x3l < n M \\ p - x U • “ 3-1 Aecordinglyf x <= A[p,S] implies that Vf(p) • f(x)l < nM Up - xl' , and so (see V*l) (5) A[p,S] S(p,nM) * But since by hypothesis S is I^-sseasurable, it follows from the Accessibility Theorem (see 111*10) that p €- A[p»S] for almost every point p £ S , and so hy (5) and I*fc* 4 we have (6) p 6 S(p,nM) for almost every p * -106- The lemma now follows from (6) and V*5* V*29* Lemma* Given f as in V*22, let E ^ 0 be a closed subset of S and assume that f is continuous on E« If f is partially of approximate E-relative b*l*d. at almost every point of E(see V*16), then there exists a sequence of sets E^ for which the following statements hold: (i) ^ <=■ E ^ c . E , (ii) Ek is closed (a fortiori I^-aeasursble), (iii) L (E - U Ev ) - 0 , n k*l (iv) f is uniformly partially Lipschitzian on each E^* Proof: For each positive integer k define (see V*22) Ejr ” E,k « 3-1 3 Then by V*23 it is clear that CL Ej ^ < - E, and by V*2l* and 1*1*8 each E^ is closed* Moreover) from V*?6 and V.27 it is clear that f is uniformly partially Lipschitzian on each E^* We now assert that (1) L (E - U Ek) - 0 . k-1 To prove (1) we first note the obvious relations ck» n n ° o (2) E - U E - E - VJ n E.v C_ O (E - KJ E 1k ) • k-1 K k-1 3-1 3-1 k-1 J By hypothesis f is partially of approximately E-relative b*l*d* at almost every point of E, and therefore application of V*25> yields the relation -107- <30 (3) L (E - [J E. ) - 0 for J - 1,..., n . n k-1 JK How (2) and (3) imply so that since 1^ is non-negative relation (1) follows* Accordingly, the proof is complete* V*30* Theorem* Let f be a real-valued L -measurable function defined on n a bounded ^-measurable subset S of R * If f is partially of approximately S-relative b*l*d. at almost every point of S, then f is of approximately S-relative b*l*d* at almost every point of S, Proof? Assign any & > 0 * Then, since by hypothesis f is Im measurable and S is a bounded L -measurable set, there exists Z1 (see 1*5*8 ) a subset E of R& for which the following statements Cl) - (U) hold. (1) E «=- S, (2) E is closed, (3) L (S - E) < e / 2 , n (U) f is continuous on E * Now, in view of (1), (2), (1*), and the hypothesis that f is partially of approximately S-relative b.l.d. at almost every point of S (see V*19) the hypotheses of V.29 are fulfilled* Accordingly, application of V*29 yields (see 1*2.6 ) a subset E of E such that the following conditions (5) - (7) hold* -108- (5) E is 1^-raeasurable, (6) Ln(E-E*) < */2 , * (7) f is uniformly partially Lipschitzian on E • From (7) and V#28 it follows that f is of approximately E*-relative b.l.d. at almost every point of E , and hence by V.0 we have that (8) f is of approximately S-relative b.l.d* at almost every point on E* • Moreover, (3) and (6) imply that (9) L (S-E*) < L (S-E) + L (E-E*) < Now, since was arbitrary, the theorem follows from (5), (8), (9), and I#2.12 • 7*31. Theorem. Let f be a real-valued Immeasurable function defined on a bounded L -measurable subset S of rf5. Then f is of approximately n S-relative bounded linear distortion at almost every point of S if and only if f is partially of approximately S-relative bounded linear distortion at almost every point of S. Prooft This merely combines V.30 and V.21 into a single statement# V#32. Definition. Let f be a real-valued Immeasurable function n defined on an L -measurable subset S of R . Then, given any subset n E ^ (2( of S and any point p * E , f is said to have an approximate E-relative total differential at p if there exist (finite) numbers -109- ,***,A such that (see 1*5*16 ) n ap 11* f(p) - f(x) - 2 A. (p^-oc^) - ______xfcE,x-*p HP-xll exists and is equal to zero* 7*33. Lenina* Under the hypotheses of 7*32, let be a given subset of S and let p be any point of E* Then f has an approximate E-relative total differential at p if and only if there exist (finite) numbers JL ,...,A such that for any assigned Proof: This follows at once from V*32 and 1*5*16 * 7*3b* Lemma* Under the hypotheses of 7*33, if f has an approximate E-relative total differential at p, then f has an approximate S-relative total differential at p* Proof* This follows at once from V.33, since (see I*JU- 4 ) / V is monotone* 7.35. Lemma* Under the hypotheses of 7*33, if p ^ ^ E and f has an E-relative total differential at p, then f has an approximate &»relative total differential at p* Proof: This follows at once from 7*32, 17*1, and 1*5* 16 • -110- V«36* Lemma* Under the hypotheses of V*33, if f has an E-relative total differential at almost every point of E, then f has an approximate E-relative total differential at almost every point of E. Proof: Since by hypothesis f has an E-relative total differential at almost every point of E there exists a subset E* of E such that (1) L (E - E* ) - 0 , n (2) f has an E-relative total differential at each point of E* • By the Density Theorem (see I*lj*3 ) we have (3) L (E - A E ) - 0 n Now since clearly E - E*0 A E (E - E*) V (E - A E ) , it follows from (1) and (3) that y E - E * a A E) < Ln (E-E*) + L j E - A E) - 0 , and so since 1^ is non-negative (It) I^(E • E* fV A E) « 0 ♦ Take any point p E*f\ A E • Then by (2) and V*3!> we see that f has an approximate E-relative total differential at p. Accordingly, in view of (U) the lemma follows* -111- V.37* Theorem* Let f be a real-valued I^-raeasurable function defined on a bounded L^-measurable subset S of Rn* If f is of approximately S-relative b«l.d. at almost every point of S, then f has an approximate S-relative total differential at almost every point of S* Proof: Let C. > 0 be assigned* Then since by hypothesis f is of approximately S-relative b«l,d* at almost every point of S there exists by V*9 a set E for which the following statements (1) - (h) hold: (1) E C. s , (2) E is closed, (3) Ifc(S-E) < t , (U) f is Lipschitzian on E* Mow from (h) and the Rademacher-Stepanoff Theorem (see37*10) it follows that (5) f has an E-relative total derivative at almost every point of E* But, in view of V*36, then f has an approximate E-relative total differential at almost every point of E and hence by V*3h (6) f has an approximate S-relative total differential at almost every point of E* Accordingly, by reason of (1),(2),(3) and (6) the theorem now follows from 1*2*12 » -112- V*38* Theorem* Let f be a real-valued L -measurable function defined n on a bounded I^-measurable subset S of B*1, If f has an approximate S-relative total differential at almost every point of S, then f is of approximately S-relative b«l.d, at almost every point of S, « Proof: By hypothesis there exists a subset S of S such that (1) Ln(S - S* ) - 0 , (2) f has an approximate S-relative total differential at each point of S* • Take any point (3) p € S* • Then, in view of (2) and V.33 there exist (finite) numbers .,A such that notation of V*l) To prove (5) take any point x S such that (6) | f(p) - f(x) - A . ^ - x ^ ) \ < v\p - x\\ ♦ 3-1 3 From the triangle inequality and (6) ve obtain \f(p) - f(x)\-nMflp-x# < |f(p) - f(x)|- 2 UU 4H P^-oc^l < > 1 3 < lf(p) - f(x) «S a 1(p 5«x ^)\ < l\p - x # , 3-1 3 so that 1 f(p) - f (x) \ < (nM+1) U p - x U • Accordingly (see V*l) , x £ S(p,nM+l) and hence (5) is established* Nov (I) and (5) imply (see l* k« k ) p ^ S(p,nM+l ), whence (see V.5) (7) f is of approximately S-relative b*l,d. at p* Thus (3) implies (7)» so that in view of (1) the theorem is proved* V*39» Theorem* Let f be a real-valued L -measurable function defined n on a bounded L -measurable subset S of H?* Then f has an approximate n S-relative total differential (see V»32) at almost every point of S if and only if f is of approximately S-relative bounded linear distortion (see V.5.) at almost every point of S, Proof: This merely confcines theorems V*37 and 7*38* - 111- V*ljO. Definition* Let f be a real—valued L —measurable function defined on an Immeasurable subset of S of Rn* Then, given any subset of S, any point p 6 E, and any positive integer j < n , f is said to have an approximate S-relative .j-partial derivative at p if there exists a (finite) number A such that (see I*5«15 and 1*5*4 ) exists and is equal to zero (where it is to be understood that the approximate limit is taken tobe w«(E A p) - relative i 3 ’ ! at pJ , whence L^-maasure is to be used) * If f has an approximate E-relative j-partial derivative at p for j • l,**.,n, then we shall say that f has approximate E-relative partial derivatives at p. V*l&. Under the hypotheses of V*IiO, let E ^ be a given subset of S , p any point of E, and j < n a positive integer* Then f has an approximate E-relative j-partial derivative at p if and only if there exists a number A such that for any assigned € > 0 pj tr E(p,A,j,fc ) , where E(p,A,j,fc) denotes the set of real numbers § such that (•) $ <•- *J(E 0 s ^wgj) , J j 3 Proof: This follows at once from V#i£> and I.5.16 • V«l(2« Lemma. Under the hypotheses of VJ:1, if f has an approximate E-relative j-partial derivative at p, then f has an approximate S-relative j-partial derivative at p. Proof: This follows at once from V.liO and 1.5.17 • V.U3. Lemma. Under the hypotheses of V.ijl, if p 6/^rr a (E 0 and f has an E-rslative j-partial derivative at p, then f has an approximate E-relative j-partial derivative at p. Proof: This follows at once from V.itO and 1.5. 18 • V.ljii. Lemma. Under the hypotheses of V.!*l , if E is L -measurable n and f has an E-relative j - partial derivative at almost every point of p, then f has an approximate E-relative j-partial derivative at almost every point of E , Proof: Since by hypothesis f has an E-relative ^-partial derivative at almost every point of E there exists a subset E^ of E such that (1) - 0 , (2) f has an E-relative j -partial derivative at each point of By the Sectional Density Theorem (see II.1 6 ) since E is by hypothesis Immeasurable, there exists a subset E2 of E such that (3) y E . e 2) - o , -116- Now (1) and (3) clearly imply (5) Ln(E - e1A e 2) - 0 • Take any point p £ E^Ae,, • Tben b7 (2),(b), and V^U3 f has an approximate E-relative j-partial derivative at p. Accordingly, in view of (5) the lernna is proved* 7*1(5* Theorem* Let f be a real-valued I^-measurable function defined on an Immeasurable subset S of Rn* If f is partially of approximately S-relative b*l.d. at almost every point of S, then f has an approximate S-relative j-partial derivative at almost every point of S for j « l,*..,n. Proofs Let 6 > 0 be assigned* Then since by hypothesis f is partially of approximately S-relative b.l*d* at almost every point in S, we know (see V*31) that f is of approximately S-relative b«l*d* at almost every point of S* Hence by V*10 there exists a set E such that the following statements (1) - (U) hold: (1) E S , (2) E is closed, (3) Ln(S - E) < € , (U) f has an E-relative total differential at almost every point of E * By 17*1 it follows from (U) that (5) f has an E-relative j-partial derivative at almost every point of E for j ■ 1,*.., n ♦ Wow V«W» and (!>) imply that f has an approximate E-relative j-partial derivative at almost every point of E, and hence in view of V*I*2 (6) f has an approximate S-relative j-partial derivative for j ■ 1,..., n at almost every point of E. Accordingly, in view of (1),(2),(3), and (6), the theorem follows from 1*2.12 * V.U6. Theorem* Let f be a real-valued I^-measurable function defined n on a bounded Immeasurable subset- S of H * If f has an approximate S-relative j-partial derivative at almost every point in S for an integer j < n , then f is j-partially of approximately b*l.d* at almost every point in S* Proof: Let j < n be a given positive integer* Since by hypothesis f has an approximate S-relative j-partial derivative at almost every point of S there exists a subset S* of S such that (1) Ln (S - S*) - 0 , (2) f has an approximate S-relative j-partial derivative at each point of S * Take any point (3) P € . s* . -118- Then in view of (2) and V#l»l there exists a (finite) nuaber A such that (10 p3£ Z\S(p,A,j,l ) , where S(p,A,j,l) has the meaning defined in V.lil • We assert that, if we set M ■ \ A\ + 1 , ve have (see V # U ) (5) S(p,M,j) -Z> S(p,A,j,l) For let S ^ S(p,A,j,l) • Then (see V#hl ) the following statements (6),(7) hold# (6) § £ " a (S^tr"1 n p ) , 3 3 3 (7) I f(p) - f(x) - A ^ - x 3)^ < lp3-x*H if S, Vjx « *^p, w*x» * Take any point i t s with tr^x ■ «^p , «j x • J • Then by the triangle inequality and (7) we have lf(p) - f(x)\ - < lf(*) - f (p) ” a(p'5-x3)1 < < \ P - x \ , and so jf(p) - f(x)[ < ( IAI-H ) I p3 « x 3 I . Accordingly, (8) | f(p) - f(x)| < M 1 p3-*3 I if x € S, tr^x ■ fljj), * x ■ § • i j j Thus, from (6) and (8) it follows that (see V. 11) 5 S(p,m,j) , and so assertion (5) holds. Now by 1.4.4 relations (4 ) and (5) imply that p^ /2i>S(p,M,j) and so by V. 16 it follows that (9) f is j-oartially of approximately S-relative b.l.d. at p. Since thus (3) implies (9), with (l) the theorem is complete. V.47. Theorem. Let f be a real-valued Ln-measurable function defined on a bounded Ln-measurable subset S of Rn. Then f has apnroximate S-relative partial derivatives (see V.40) at almost every point of S if and only if f is partially of aporoximately bounded linear distortion (see V.16) at almost every point of S. Proof: This merely combines theorems V.45 and V.4 6 . V.48. STEPANOFF THEOREM. Let f be a real-valued Immeasurable function defined on a bounded Ln~measurable subset S of Rn . Then f ha.s an apnroximate S-relative total differential at almost every point of S if and only if f has aproximate S-relative par tial derivatives at almost every point of S. Proof: This follows at once from V.39 and V.47. -120- Bibliography 1* Caratheodory, C. Vorlesungen ueber reele Funotionen, Leipzig- Berlin, 2 Aufl. 1927. 2. Graves, L. M. The Theory of Functions of Real Variables, McGraw-I3.il, New York-London, 1946. 3. Kuratowski, C. Topologle I, Warsawa-Lwow, 1933. 4. Mayrhofer, K. Inhalt und Mass, Springer, Leipzig, 1952. 5. McShane, E. J. (a) Integration, Princeton University Press, Princeton, 1947; (b) Extension of Range of Functions, Bui. Amer. Math. Soc., Vol. 40 (1934), pp. 837 - 842. 6. Rademacher, H. Peber partielle und totals Differenzierbarkeit I, Math. Ann., Vol. 79 (1919), pp. 340 - 359. 7. Rado, T. On absolutely continuous transformations in the plane, Duke Math. Journal, Vol. IV (1938), pp. 219 - 220. 8. Saks, S. Theory of the Integral, Warsawa-Inrow, 1937. 9. Stepanoff, W. Sur les conditions de 1 1existence de la differentielle totale, Recueil Math. Soc. Math. Moscou, Vol. 32, pp. 511 - 526. -121- AUTOBIOGRAPHY I, Albert George Fadell, was born in Niagara Falls, New York, January 5, 1928. I received my secondary school education in the public schools of the city of Niagara Falls, New York. My undergraduate training was obtained at the University of Buffalo, from which I received the degree Bachelor of Arts in 1949 and the degree Master of Arts in 1951. ifVhile completing my requirements for the degree Master of Arts I acted in the capacity of Teaching Fellow at the University of Buffalo. In 1951 I received an appointment as Graduate Assistant in the Department of Mathematics of the Ohio State University, and held this position for three years while completing my require ment for the degree Doctor of Philosophy. -122- 0 be any assigned number, such that the following conditions holdt