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.