I
71-27,463
ENGLE, Jessie Ann Nelson, 1918- HAAR MEASURE ON LEFT-CONTINUOUS GROUPS AND A RELATED UNIQUENESS THEOREM.
The Ohio State University, Ph.D., 1971 Mathematics
University Microfilms, A XEROX Company, Ann Arbor, Michigan
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED HAAR MEASURE ON LEFT-CONTINUOUS GROUPS
AND A RELATED UNIQUENESS THEOREM
DISSERTATION
Presented in Partial Fulfillment of the Requirement for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Jessie Ann Nelson Engle, B.A., M.S.
* x- * * *
The Ohio State University 1971
Approved by
9- Adviser Department of Mathematics ACKNOWLEDGMENT
The writer wishes to express her deep appreciation to her dissertation adviser* Professor Earl J. Mickle, for suggesting this topic for research; for his continued advice and aid in connection with it; and for his help and guidance throughout all the writer's years in graduate school.
ii VITA
September 17, 191$ • ...... Born - Chicago, Illinois
1940...... B.A., Bennington College, Bennington, Vermont
1940-1942. • ...... Music Teacher, Black Mountain College, Black Mountain, North Carolina
1942-1944. * • ...... Apprentice Technician, Columbia Broadcasting System, New York, New York
1944-1945...... Research Associate, Radio Research Laboratory, Harvard University, Cambridge, Massachusetts
1946-1947...... Research Associate, Airborne Instrument Laboratory, Mineola, New York
1947-1948...... Graduate Assistant, Electrical Engineering Department, Stanford University, Stanford, California
1948 ...... Research Engineer, Done and Margolin, Bethpage, Lew York
1949-1953...... Recording Studio Technician, Juilliard School of Music, New York, New York
1957-1963...... Viola Player, Columbus Symphony, Columbus, Ohio
1963-1971...... Teaching Assistant, Mathematics Department, The Ohio State University, Columbus, Ohio
1964 ...... M.S., The Ohio State University, Columbus, Ohio
iii PUBLICATIONS, ETC.
"Ultrahigh-Frequency Measurements," (Jessie A. Nelson, David Lazarus, John W. Christenson, and Robert R. Buss); "Impedance Matching, Transformers and Baiuns,” (Jessie A. Nelson and Gus Stavis); and "Sleeve Antennas," (Ernest L. Bock, Jessie A. Nelson, and Arthur Dorne); all from Very High-Frequency Techniques, compiled by the staff of the Radio Research Laboratory, Harvard University. New York. McGraw-Hill. 19^7•
"Antenna System," (to Airborne Instrument Laboratory, Inc., Mineola, New York), U. S. Patent 2 ,555,857, 5 June 1951* (Jessie A. Nelson and Arthur Dorne).
"Broad Band Antenna," (to the United States of America as represented by the Secretary of War), U. S. Patent 2}62h,8hk, 6 January 1953- (Jessie A. Nelson and Darrel I. Wilhoit).
"Broad Band Antenna System," (to the United States of America as represented by the Secretary of War), U. S. Patent 2 ,633,531, 31 March 1953. (Jessie A. Nelson).
"Coaxial-Line Switch," (to the United States of America as represented by the Secretary of War), U. S. Patent 2,662,1^12, 8 December 1953» (Jessie A. Nelson).
FIELDS OF STUDY
Major Field: Mathematics
Studies in Analysis. Professors B. Baishanski, F. Carroll, R. Helsel, L. Meyers, and E. J. Mickle.
Studies in Probability. Professor L. Sucheston.
Studies in Topology. Professors N. Levine, E. J. Mickle, and P. Reichelderfer. TABLE OF CONTENTS
Page
ACKNOWLEDGMENT...... ii
VITA ...... iii
LIST OF SYMBOLS...... vi
INTRODUCTION...... 1
Chapter
I. THE EXISTENCE OF A LEFT-INVARIANT MEASURE ON A LEFT-CONTINUOUS GR O U P ... 6
II. CONDITIONS EQUIVALENT TO CONDITION (i) . 21
III. AN EMBEDDING THEOREM...... 29
IV. A FIVE-R UNIQUENESS T H E O R E M ..... 38
V. AN EXAMPLE...... 65
BIBLIOGRAPHY...... 7^
v LIST OF SYMBOLS
a,bjX ...... elements of a set
A j B , X ...... subsets of a set
A,T,F ...... families of subsets of a set * U ...... a uniformity on a set
U ...... a set in a uniformity
AB ...... {ab: a 6 A, b 6 B) a B ...... {ab: b 6 B] c l A ...... the closure of A i n t A ...... the interior of A
V"1 ...... {v"1 : v 6 V}
vi INTRODUCTION
The locally compact topological group is a structure which
generalizes certain properties of the real line, among them the transla-tion-invariance of Lebesgue measure and the continuity of translations with respect to the euclidean topology. The Haar measure which exists on any locally compact topological group
is a generalization of Lebesgue measure on the line.
This dissertation is concerned with determining sufficient
conditions for the existence of a non-trivial invariant measure on a structure which is weaker than a locally compact topological
group. The weaker structure chosen for study is a group endowed with a locally compact Hausdorff topology with respect to which •
ail left translations are continuous. Such a group is here called a "left-continuous group."
Only left-invariance and left translations are considered,
since the theory for right-invariance and right translations is
symmetrical.
In Chapter I it is shown that there exists a non-trivial left- translation-invariant measure on any left-continuous group which
satisfies Condition (i), a condition on the open neighborhoods of the group identity.
A left-continuous group; a left-continuous group which satisfies
1 2
Condition (i); and a locally compact Hausdorff topological group— these are successively less general structures. The relationship among the three is made clear when a neighborhood characterization of a topological group is used, as in Bourbaki [3 ]. (Numbers in square brackets refer to numbered entries in the bibliography.)
Common to all three is an underlying structure, a group G with a locally compact Hausdorff topology. We let U be the family of open neighborhoods of the identity in G, and consider the following conditions which may be imposed on the underlying structure.
i) For every x in G and every open set 0, the set
xO is an open set.
i 1) For every x in G and every open set 0, the sets
xO and Ox are open sets,
ii) For every U in U, there is a V in U with c U.
iii) For every U in TJ, there is a V in U with V ^ c U.
iii') For every U in U, is in U.
iv) For every x in G and every U in U, the set xUx"’*' is
in U.
If condition i) is satisfied, the structure is a left-continuous group; if conditions i), ii), and iii) are satisfied, the structure is a left-continuous group which satisfies Condition (i); and if conditions i ’)> ii), iii1)* and iv) are satisfied, the structure is a locally compact Hausdorff topological group.
The main result in Chapter I is that Haar measure exists on a structure"”that is substantially weaker than a topological group, 3 that is, that Haar measure exists on a left-continuous group which satisfies Condition (I).
Chapter II discusses equivalent conditions for the existence of a Haar measure on a left-continuous group.
A left-continuous group may be regarded as a locally compact
Hausdorff topological space (X, T) operated on by the family H of left translations, x -» ax. The translation associated with each a in X is a homeomorphism from X onto X; the family H is a group under composition; and H is transitive, that is, for each pair of elements x,y in G, there is an element h in H associated with Or a = yx”1 such that y = h (x). 3» It is shown in Chapter II that imposing Condition (i) on a left-continuous group is equivalent to imposing certain conditions on the family H. Equivalent conditions are l) that the topology on X be a uniform topology and that the family H be equicontinuous with respect to a uniformity which induces the topology; 2 ) that the family H be strongly evenly continuous; and 3) that Strong
Condition A hold: for every pair of disjoint sets, one closed and one compact, there is an open set such that no left translate of the open set intersects "both the closed set and the compact set.
In Chapter III a method is given for constructing left- continuous groups from topological groups. A topological group may be embedded in a left-continuous group, and •under certain conditions the embedding group will not be a topological group. For instance, if for some x in G, the set [a: axa”"*- = x} is not an open set, then k there is an embedding left-continuous group which is not a topological group. In this case the embedding group is a product space with a non-normal subgroup which is isomorphic to the embedded topological group. Left cosets are copies of the embedded group, and are endowed with a copy of the topology on the embedded group. Since right cosets are not unions of open sets, right translations are not continuous, and the embedding group is not a topological group.
Chapter IV contains a related uniqueness theorem— a 5-r uniqueness theorem for invariant measures on a separable metric space. The theorem is a sharpening of the 5-r uniqueness theorem of Mickle and
Rado [8] which was proved by them to show the uniqueness of a translation-invariant measure on the space of oriented lines in three-space. The measure under consideration is Haar with respect to a family of homeomorphisms which satisfy Condition (II), a condition guaranteeing that, translates of closed spheres of small radius are bounded within and without by closed spheres of related radius. It is shown in Chapter IV that a Haar measure is unique, up to a multiplicative constant, when it satisfies the 5-r condition at one point. The 5-r condition is a condition on the rate of increase of measure of a closed sphere of arbitrarily small radius as the radius increases.
The final chapter, Chapter V, contains an example of a left- continuous group with its associated left-translation-invariant measure. The group is the open right half-plane, that is, the product space C(x,y): x > 0, y is real}. The group operation is 5 defined by (x,y)(a,b) = (xa, xb + y). Open sets are unions of sets which are open intervals on horizontal lines; e.g.,
C(x,yQ): a < x < b} is an open set. With this topology, left translates of open sets are open, and right translates of open sets are open if and only if the translating element is of the form
(a,0). This example illustrates the embedding theorem: the subgroup
{(x,0 ): x > 0 } is a topological group with the induced subspace topology, and is an open neighborhood of (1,0), the identity in G.
Hence Condition (l) is satisfied, and there is a Haar measure on the left-continuous group, a measure A which on open neighborhoods of (x , y ) takes the value 0 0
A[{(x, yQ): xQ € (a,b), a < x < b}] = In ~ , where In is the natural logarithm. The measure A is clearly non-trivial and left-translation-invariant. CHAPTER I
THE EXISTENCE OF A LEFT-INVARIANT MEASURE
ON A LEFT-CONTINUOUS GROUP
This chapter is concerned with defining a non-trivial left-
translation-invariant measure on a left-continuous group. A left-
continuous group is a group, with the group operation here called
multiplication, which is endowed with a locally compact Hausdorff
topology with the property that all left translations are continuous, where left translation by an element x in G is multiplication on
the left by x.
Clearly every locally compact Hausdorff topological group is
a left-continuous group, but there are left-continuous groups which
are not topological groups, and an example will be given below in
Chapter V. It was proved by Ellis [U] in 1957 that the continuity
of both left and right translations, or continuity of either the
left or right translations and of the inverse operation, guarantee
that a group with a locally compact topology be a topological group.
Thus locally compact semi-topological groups, para-topological groups,
and quasi-topological groups, to use Bourbaki ([3 ]) nomenclature,
are all topological groups, whereas a left-continuous group is
locally compact, but not necessarily a topological group.
6 7
Definition 1.1.
A left-continuous group is a structure (G,» ,T) where (G,o) is a group, (G, T) is a locally compact Hausdorff topological space, and for every element g in G and every open set 0 in T the set gO = {gx: x € 0} is an open set.
Definition 1.2.
A left-continuous group (G, ® ,T) is said tosatisfy Condition
(i) if for every U, with e € U .€ T (where e is the group identity in G), there is a V, with e € V € T such that W ^ c U.
Note that the set v”^ is not necessarily an open set.
Definition 1.3» r
A non-negative real-valued set function A defined on all subsets of a set G will be said to be an outer measure on G if
i) A(d>) =0, where (J) is the empty set,
ii) for A c B, A(A) < A(b ), and
iii) for An, n = 1,2,***, a sequence of sets in G,
Definition 1.4.
An outer measure A is said to be Borel-regular if for every set E in G there is a Borel set B such that E C B and A(E) = A(B).
Theorem 1.5. (Existence Theorem).
If (G,o, T) is a left-continuous group satisfying Condition (I), 8 then there exists an outer measure A on G such that
1) A is Borel-regular,
2) if E c G, x € G, then A(xE) = A(E),
3) if 0 € T, 0 1 (j), then A(0) > 0,
J+) if K is compact, then A(K) < <=, and
5) Borel sets are A-measurable.
The proof of Theorem 1.5 will follow after a series of lemmas.
For the remainder of Chapter I, (G, 0 ,T) is a left-continuous group which satisfies Condition (i). Several consequences of assuming
Condition (i) are given in Lemmas 1.6 through 1.11. Since the topology on G is locally compact and Hausdorff, it is also regular, and each point in G has a closed neighborhood base. Below, U e will denote the family of open neighborhoods of the identity element e in G, and £ will denote the family of closed neighborhoods of e.
Lemma 1.6.
For every U in U there is a V in U such that V2 c U. — e — e
Proof:
Fix U € U . Then there is W 6 U with c U, and — e — e V € U e with W _1 c W, and V2 c w “1W "1 = W -1 (W "1 )"1 cwvf1 c U.
Lemma 1.7«
For every C in C there is a D in C such that DD ^ c C, — e —e
Proof:
Fix C € C . Then there is U € U with IJC C, V S U with — e — e — e 9
W ' 1 c U, and D 6 C with D c V, and Dlf1 c w "1 c U c C. — e
Lemma 1.8.
For every compact set K and open set 0 that contains K, there is a U in U such that KU c 0. — e
Proof:
For each x f K c 0, there is a W € U with xW c o, and there X "* 6 X is a V G U with c W . Since K is compact, and K <=U{xV : x S K], x -e xx x ' there is a finite set, {x,, x5,••*, x } such that K c Q x.V . L u n • > i x* i=l i n Letting Y = 0 V * we have V 6 U and i=l Xi "'e
n n n K V C Ux.Vv Vcr Ux V V c U x.W c 0. i=l 1 i i=l 1 i i i=l 1 i
Lemma 1.9.
For every subset A of G, and every open set 0, the set AO is an open set.
Proof:
If x € A, then xO € T, and
AO = U {xO: x € A) € T.
Lemma 1.10.
For every compact set Kand closed set F where K and F are disjoint, there is a U in U g such that KU and FU are disjoint.
Proof:
It suffices to show there is a U 6 U g such that KUU ^ H F = 4, 10 since if KU fl FU 7^ <}), we have = f^n^ for some u^, Ug in U and kx in K, f2 in F. Then fg = k ^ U g "1 € KUU*1 0 F ^
Since K fl F * I, K c (g > F)6 T, and by Lemma1.8 there is a V € U such that KV cr (g - F). Then there is a U in U with — e — e UU*1 C V, and since KUU*1 CKVC (G - F), we have KUU-1 fl F = Lemma 1.11. For every compact set K and closed set F with K and F disjoint, there is a U in U g such that no left translate of U has non-enrpty intersections with both K and F. Proof: Let K and F be disjoint, and compact and closed respectively. Then, by Lemma 1.10, there is a V f U g such that KV 0 FV = (UU*1 )2 c: v. Assume xU fl K f Then fg = xUg=k1u1"lu2 C ^ U ^ U c k^UU *1 )2 c KV, and fg € KV n F C KV n FV a <|). Contradiction. Hence xU (1 F = |). Below, in Chapter II, there is a discussion of various conditions which are equivalent to Condition (i). The following three lemmas lead directly to the proof of Theorem 1.5* the existence theorem. In what follows, (G, 0 , T) is a left-continuous group which satisfies Condition (i), U . is the ** 6 family of open neighborhoods of the identity in G, and F will denote 11 the class of compact sets in G with non-empty interiors. Since (G, o ,T) is locally compact, every element in G has a neighborhood base of sets in F. In Lemma 1.12, a left-invariant real-valued set function X will be defined on sets in F. Lemma 1.12. There is a real-valued set function X defined on the family F of compact sets with non-empty interiors such that for all A,B in F and all x in G: 1) 0 < X(A) < 2) if A c B, then X(A) < X(B), 3) X(A UB)< X(A) + X(B), 4) X(A) = X ( x A), and 5) if A fl B 4 , 'then X(A U B ) = X(A) + X(B). Proof; For A,B 6 F, let n (A:B) = inf{n: A cr U x. intB}, i=l 1 where intB is the interior of B. Then for A,B,C € F and for x € G, we have; i) 1 < (A:B) < «, ii) (xA:B) = (A:B) = (A:xB), iii) if A <= b , then (A:C) < (B:C), iv) (A:C) < (A:B)(B:C), V) (A U B;C) < (A:C) + (B:C), and vi) if A (1 B = 1.11. For A,B € F, A 0 B = (B:K) < (A U B:K), and equality follows. Fix Fq € F, and define, for each A,B € F, Then for all A,B,C 6 F, and x € G, we have: (i) V A) =3 >-b(xA)> (ii) if A c C, then Xfi(A) < X£(C), (iii) VAU c) < xb (a ) + xB(c), (iv) if A n C = 4>, then there is B € F with xB(A u C) = Xb (A) + XB(C), and (v) 0 < TTT a J S V * ) < (A:F0 ) < ' O ' Conclusions (i) through (iv) are obvious, and (v) follows from iv) above: 13 From (v), it can be seen that for each A in F, the set {\_.(A): B 6 F) is a subset of the closed interval [l/(F :A), (A:F )] , J3 O O which is a subset of (0 , “). Let P be the product space, indexed by {A: A € F], of the closed intervals [1/(Fq :A), (A:Fq )], with the product topology induced by the usual real-line topology on the closed intervals: p - fT [ u h a > • A € F L ° Since each interval is compact, the product space P is compact (by the Tychonoff product theorem; see Kelley [7])> and for each B in F, Xg is an element of the product space P. For each compact neighborhood K of the identity element in G, that is, for K 6 —C e O F — , let S(K) = {X-. B € C n F, B c K}. Sj “ C — Then S(K) c P for each K, and clS(K), the closure of S(K), is a closed subset of the compact space P, hence is compact. The family of clS(K) for K ? C O F has the finite intersection property, since n C1S(K. ) 3 cl n n S(K. ) = cl nn (x«: B€c OF,BCK.) i=l 1 i=l 1 i=l * 6 n = clfX^: B 6 C OF, BC pK. 6 FflC } ^ . Thus, 0 (clS(K): KGFnCg}^. Let X € 0 (clS(K): K € FflC^. Then X is a real-valued function on F that satisfies (l) - (5) of the lemma. 1U It is clear that (l) holds, since X € P and for every A in F, 0 < 1/(Fq :A) < X(A) < (A:Fq) < °>. For an arbitrary element q in P and set A in F, let p^( q) be the projection of q at A. Then p^ is continuous for all A in F. The projections pA are used in the proofs of (2) - (5). To show that (2) holds, let A C B and let ^ = Cq: PA(q) < Pfi(q)}. Since pA and p^ are continuous, ^ is a closed set, and for K € C g n F, XR € 0^ since Xr (A) < Xk (B). Hence cIS(k ) <= and X 6 Then if A c B, X(A) < X(B), and (2) holds. Similarly, to show that (3 ) holds, for A,B 6 F, let Qg = P(aU b ) W < + 111611 % is closed* clS(K) <= ^ and X € 0^, hence X(A U B) < X(A) + X(b ). For arbitrary A € F and x f G, let Qij. = Cq-* PA(q) ^ p ^ C q ) } - Again clS(K) c q ^, X € Q^, X(A) = X(xA), and (U) holds. To establish (5), let A,B be disjoint sets in F. Let 05 = Cq: p a y B (q) = pA (q) + pB (q))* Then is a closed set. Now there is a K in F such that Xr (A U B) = Xk (A) + >-k (b ) (by (iv) above), and by ii), for x € intK, Xx-ik (A UB)= + xx-lK(B)a x_1K € C e . Then for D c x_1K, D ecg, we have Xd(A U B) = Xd(A) + Xd(b), and S t x ^ J c Q ^ , clS(x"'1K) c q^, and X € Q^. Hence X(A (J B),= X(A) + X(b) when A fl B = The real-valued set function X, defined on the family F of compact sets with non-empty interiors, will be used to define a set f-unction p on all open sets in G. Lemma 1.13. For all 0 in T, the family of open sets in G, let p(0) = sup{X(F): F € F, FCO] for 0 4 (J), and p(6) = 0. Then for A,B in T, x in G: (1) if A c B, then p(A) < p(B), (2) p(A U B) < p(A) + p(B), (3) p(A) = p(xA), (h) if A H B = Proof: For every 0^ Conclusions (l), (3), and (U) are obvious from the definition of p and the properties of the set function X. To prove (2), let FCAU B, F € F. Then F c U {0 : x 6 A, x € 0 € T, clO c A} A A A u U {0y : y 6 B, y € 0y € T, clOy c B% n m Since F is compact, F c: U 0 U U o for some (x.), i=l,2,»»»,n, i=l i i=l yi 16 and (y\), i=CL,2,**»,m, and letting n m F = F n U cio , F« = P n U clO , X i=l xi * i=l yi we have F^,Fg f |, F^ c A, Fg B, and F^ U Fg = F. Then X(F) < XC*^) + ^(F2^ — + since F 1'*a -s 811 arbitrary subset of A U B, we have p(A U B) < p(A) + p(B). 00 To prove (5), first assume p( U 0. < Then for every e > 0, i=l 1 CO 00 there is an F in F with F C (J 0- > X(f) > p( U 0. ) - e. Since F i=l 1 i=l 1 is compact, there are a finite number of the CL’s that cover F: n F C IJ 0- • Then i=l x co n n 00 p( U 0. ) - e < X(F) < P( U 0 . ) < 2 P(0. ) < 2 p(0. ), i=l 1 i=l i=l i^L where the second inequality follows from (2). Since e is arbitrary, p( U O ) < Z p(0 ) 1=4 i=l 00 when p( U 0 .) < i=l CO Assuming p( U 0.) = <», we have for every n an Fn in F with i=l 1 CO n < X(F ") and P c U (0.). Since F is compact, there are a finite ' n' n . , i' n s ’ i=l m number of the 0.'s that cover F : F_ c: IJ (0.), and 1 i=l 1 m m » n < *(*■ ) < p( U 0 ) < z p(o.) < z p(oi ). i=l i=l i=l 17 0 0 Since this is true for arbitrary n, 2 p(0.) = «*, and (5) holds. i=l 1 Definition l.lfr. For each subset E of G, define A(E) = inf{p(o): 0 € T, E c 0}. Lemma For every open set 0 in T, A(0) = p(0). Proof: Let 0 6 T. Then since 0 c 0, A(0) < p(0). For every Q in T with 0 c Q, we have p(0) < o(q), so p(0) < inf{p(Q): O c Q, Q € T} = A(0), and A(0) = p(0). The existence theorem, Theorem 1.5* will be restated and proved, vising the preceding lemmas. Existence Theorem. If (G, o ,T) is a left-continuous group satisfying Condition (i), then there exists an outer measure A on G such that: 1) A is Borel-regular, 2) if E <= G, x € G, then A(xE) =A(E), 3 ) if 0 € T and 0 ^ 6, then A(0) > 0, I*) if K is compact, then A(K) < and 5) Borel sets are A-measurable. Proof: Let A be the set function defined in Definition l.lU. Then A 18 is an outer measure if A( 00 0 0 If (A.). , is a sequence of subsets of G, and Z A(A.) = <», x i-J. i= 1 x CO 00 00 then A( U A. ) < Z A(A. ). So assume Z A(A.) < “>. Then A(A.) < “ i=l i=l i=l for each i. Fix e > 0. For each A^ there is an CL with A^ c (L CO 00 00 and p(0.) < A(A. ) + e/21, and A( U A.) < p( U O ) < Z p(0 ) < 1=1 1=1 1=1 00 Z A(a.) + e. Hence A is countably subadditive, and A is an outer i=l 1 measure. The fact that A is Borel-regular follows from the definition of A. For arbitrary E c G, if A(E) = 00, then E c G, A(G) = », and G is in T, hence a Borel set. If A(E) < ra, for each n there is an 00 0 containing E with A(e ) + l/n > o(0 ). Then H 0 is a Borel n n n=l n 00 set which contains E, and A( H 0 ) < p(0 ) < A(E) + l/n for every ' n=l n n CO 00 n, so A( (1 0 ) < A(E). Since any open set containing 0 0 also n=l n n=l 00 00 contains E, A(E) < A( Pi 0 ), and A(E) = A( 0 0 ). n=l n n=l n Conclusion 2) follows directly from Lemma 1.13 (3). To prove conclusion 3)j let 0 be an arbitrary non-empty open set> and x a point in 0. There is a closed neighborhood C of x such that C c o, and a compact neighborhood K of x, since the topology is regular and locally compact. Then C PI K is in F, and X(C fl K) > 0, by Lemma 1.12, l), so A(o) = p(0)>\(CPK)> 0, and A(0) > 0. 19 Conclusion 4), that for every compact set K, A(K) < °°, follows from the compactness of K and the left-translation invariance of A if it can be shcam that there is one non-empty open set with finite A- measure. Fix x, and let C be a compact neighborhood of x. Then intC is an open neighborhood of x, hence non-empty, and 0 < A(intC) = sup{X(F): F c intC, F € F} < X(C) < ro. Any compact set K may be covered by a finite number of translates of intC, each of finite A-measure; hence A(K) < °°. To prove that all Borel sets are A-measurable, it suffices to prove that all open sets are measurable, and to prove all open sets are measurable, it suffices to show, for arbitrary 0 in T and E c g with A(E) < °°, that A(E) > A(E n 0 ) + A[E D (G - 0 )]. The notation of Lemmas 1.12 and 1.13 is used below. Let Q- be inT with E c q and A(Q) <“. Then 0 fl E e (0 '0 Q) € T, and E 0 (G - 0) c Q n (G - 0). For every n there is an Fn in F such that Fn C 0 n Q, A(0 n Q) = p(0 n q) < X(Fn) + l/n, and there is an 0 in T and a K in F such that n — n — E n (G - 0) C on C q n (G - Fu^ Kn cr on, and A(0n ) = p(0n ) < X(Kn ) + l/n. Then A[E 0 (G - 0)] + A(E fl 0) < A(On ) + A(0 0 Q) < X(Kn ) + l/n + X(Fn) + l/n = X(Kn U Fn ) + 2/n < p(Q) + 2/n = A(Q) + 2 /n. The first equality holds since H Fn = a[e n (g - o )] + a(e n o) < a(q) + 2 /n holds for every n, we have A[E H (G - 0)] + A(E fl 0) < A(q ) for every open set Q containing E, hence a[e n (g - 0 )] + a(e n o ) < a(e). The proof of the theorem is complete. An example of a left-continuous group which satisfies Condition (ii and the left-invariant measure that exists on it, will be given below in Chapter V. CHAPTER II CONDITIONS EQUIVALENT TO CONDITION (i) In Chapter I it was shown that if a left-continuous group G satisfies Condition (i) there exists a non-trivial measure on G that is invariant with respect to left translations. In this chapter some conditions equivalent to Condition (i) will he given. Definition 2.1. A left-continuous group (G, ° ,T) will be said to satisfy Condition (la) if for every U in U (where U is the family L 6 1 * © of open neighborhoods of the identity in G) there is a V in U fi such that V2 c u. Definition 2.2. A left-continuous group (G,° ,T) will be said to satisfy Condition (lb) if for every U in U g there is a V in U 0 such that V- 1 c u. Lemma 2.3» A left-continuous group (G,0 ,T) satisfies Condition (i) if and only if it satisfies Conditions (la) and (lb). Proof: Assume Condition (i) holds, and let U be an open set in 'U w . 21 22 Then there is an open set W in U g with ww"1 c: U and there is an open set V in U with -l W c W. Now W = VeVe c w " 1W - 1 = ( w ”1 ) (w"1 )"1 c WW- 1 c U, and V-1 = eV*1 c w "1 c w c WW-1 c u. So if Condition (i) holds, Conditions (la) and (lb) hold. Assume Conditions (la) and (lb) hold, and let U be an open set in U . Then there are sets W and R in U with — e — e w2 c u, r”1 c w. Letting V = R 0 W, we have V € U and " v w " 1 C Vffi"1 c w2 c u, and the lemma is proved. A left-continuous group (G, o ,t) may be regarded as a locally compact Hausdorff topological space (G,T) and a family H of homeomorphisms of G onto G with (l) H = (h : x € G, and for y € G, h (y) = xy}. X X Thus H is the family of left translates fron* G onto G. Each hx 23 in H is both continuous and open, and clearly H is a group with composition as its binary operation. In what follows, a left-continuous group (G, 0 ,T) and a locally compact Hausdorff topological space (G,T) with a family H of homeomorphisms as at (l) will be used interchangeably as convenient. Requiring that (G, 0 ,T) satisfy Conditions (i), (la),or (lb) is equivalent to imposing certain conditions on the family of homeomorphisms H. Steinlage [15] has defined Strong Condition A on a group of homeomorphisms from a topological space onto itself. Using the notation of this dissertation, Strong Conditions A may be defined as follows. Definition 2,k. (Steinlage) A group H of homeomorphisms of a topological space G onto itself is said to satisfy Strong Condition A if for each pair of disjoint sets B and C, one of which is compact and one of which is closed, there exists a non-empty open set 0 such that h(0) OB =<}) or h(0) fl C = (!) for all h in H. Several other conditions are also equivalent; after defining them proof of the equivalences will be given. Definition 2.% A group H of homeomorphisms of a topological space G is said to be evenly continuous if, for each pair of elements x,y in G, and for each open neighborhood 0 of y, there are open neighborhoods 0^ and Og of x and y respectively such that for every h € H, 2k if h(x) € Op, then h(0 ) c= 0 . c. x y Definition 2.6. A group H of homeomorphisms of a topological space G will be said to be strongly evenly continuous if, for every pair of elements x,y in G, and every open neighborhood 0 of y, there are open «y neighborhoods 0^ and 0g of x and y respectively such that for every h € H if h(O^) 0 0g ± 6, then h(01) c 0y . One equivalent condition concerns uniformities: Condition (I) holds on the left-continuous group (G, ° ,T) if and only if there is a uniformity on G such that the topology on G is generated by that uniformity, and the family H is equicontinuous with respect to the -uniformity. For definitions, notation, and discussion of uniformities, the reader is referred to Kelley [7]. Definition 2.7. * A family H of transformations of a uniform space (G,U) is * * said to be equicontinuous if for every x in G and every U in U there is an open neighborhood 0 of x such that for every h in H, h(0 ) e U[h(x)]. Theorem 2.8. The left-continuous group (G,° ,T) satisfies Condition (la) if and only if the family H is evenly continuous. Proof: When H is a family of left translates of a left-continuous group, even continuity of H means that for every x,y in G and 25 U in U , there are R, S in U such that if zx € yR, then zxS c yu. Assume that H is evenly continuous. Let x = y = e and fix U 6 U . Then there are R,S in U such that - e - e z 6 R =» zS o u. Let V = R 0 S. Then V € U , V2 c U zS c U, e z € r and Condition (la) is satisfied. Assume Condition (la) is satisfied. Fix x, y, and U. There is a V in U— e such that V2 c U. Let R =S = V. Then zx € yR = yV => zxS c y W c: yu, and the theorem is proved. Theorem 2.9. In a left-continuous group (G, ° ,T) -where H is the family of left translations, the following conditions are equivalent: a) Condition (i) is satisfied. b) H is strongly evenly continuous. c) H satisfies Strong Condition A. d) There is a uniformity *U on G which generates the topology * T, and H is equicontinuous with respect to U. Proof: The equivalence will be shown by proving that a) =» d) =* b) => c) =* a). * First assume Condition (i) holds. Let B be the family of sets 26 of the form U = {(x,y): x“V € U, U € U e ). * The family B is a base for a uniformity if it satisfies four conditions: (See Kelley [7].) 1) For every x inG and every U in B, (x,x) € U. 2) For every U in B, there is a V in B such that V c u * * * * * * * 3) For every U in B, there is a V in B such that V«Vcu, * * * * U) For every U and V in B, there is a W in B such that * * « * w <= u n v. Clearly condition l) is satisfied. To prove condition 2), "X* 1 let U = {(x,y): x' y f U, U f U }. Then there is a V in U such mam 0 - Q that W _1 c u. Let V = {(t,w): t-1w 6 V] = {(t,w): w-1t 6 V-1 -1 i * - 1 C w c u} c u . For 3), let U be given with the associated U € U g . There is a V in U 0 such that V2 c U. Then Vo V = {(x,y): (x,z),(z,y) € V for some z € g} = {(x,y): x-1 z, z-1y E V] = {(x,y): x V = x”1zz'*1y 6 V2 c u} c U. -X ■x* To prove k), fix U and V. Then U H V = C(x,y): x_1y € U, x *y € V} = {(x,y): x V € U 0 V] * * = (U n V) € B. * ■* Thus B is a base for a uniformity U. * * The uniformity U generates the topology T if each set U[x] contains an open neighborhood of x, and each open neighborhood of * * * x contains a set U[x] for some U 6 B. This is easily shown: 27 U[x] = fy: (x,y) € U} = {y: x-1y 6 U} = {y: y € xU} = xU. It remains to show that the family of homeomorphisms H is * equicontinuous with respect to U, that is, that for each x in G, * each U in U there is a V in U such that for every z in G, * zxV c: U[zx], This is trivially true, since U[zx] = zxU. Second, to show that d) implies b), assume there is a uniformity * U on G which generates the topology T, and assume H is equicontinuous * with respect to U. For fixed x,y, and U in U we need to show there are W and T in U such that for all z in G, if zxW fl yT £ V o V o V[y ] c yU, and by equicontinuity there is a W in T such that for every z in G, •x* •x* zxW c v[zx]. Assume t € zxW 0 V[y], and let p 6 xW. Then, since V = V ”3', (y,t) € V, (zx,zp) € V, and (zx,t) € V, we have * * ■X’ -X* *x* (y,zp) € V o V u V, and zp € V ° V » V[y] c yU for every p in xW, hence zxW c yU. Third, to show that b) implies c), fix C compact and A closed with C fl A = |). For x = e,and any y € C c (G - A) € T, bystrong even continuity there is a W such that for all z in G, if y zW fl yW 4 then zW c (G - A). Since C is compact and y y y e c u {yW : y€ c ] , there is a finite subfamily, y,W jYoK, >* * * > 7 > y " i * 2 n * 28 n n with C c Uy.W • Let W = Ow € T. Then if zW n C 4 <|), i=l 1 yi i=l yi zW H y.W r 6 for some i and zW cr (g - A), or zW 0 A = d). and y± ax yi Y Strong Condition A is satisfied. It remains to show that c) implies a), that is, that if H satisfies Strong Condition A, (G, o ,T) satisfies Condition (i). Assume H satisfies Strong Condition A, and let U € U e. We may assume that U has compact closure, since G has a locally compact topology. There is a C in C with ecu. Then C is compact, B = G - U i^ closed, and C H B = d). Hence by Strong Condition A there is a non-empty open set 0 such that xOflC=(|orxOnB=|) for every left translate in H. Choose x in 0. Then x ^0 6Ue< Let V = x-10 fl intC. Then V € U , V ^ 6, andfor v €V, wehave vV fl C ^ 4, hence vV 0 B = V2 = IJ w c u . v€ V Now for v € V, e = v^v€v "S/, and v ' V (1C 4(j>, hence v fl B = Thus if Strong Condition A holds, then both Condition (la) and Condition (lb) hold, hence by Lemma 2.3, Condition (i) holds, and the theorem is proved. CHAFTER III AN EMBEDDING THEOREM Under some conditions it is possible to embed a topological group in a left-continuous group which is not a topological group. In this chapter an embedding theorem will be stated and proved. Definition 3«1. A topological group (H,*,S) is said to be embedded in a left- continuous group (G, o ,T) if there is a group A which is a subgroup of G, and a map from A onto H which is a group isomorphism and is also a topological homeomorphism with respect to the induced sub space topology on A. Theorem 3.2. A topological group (H,•,S) may be embedded as a proper subgroup in a left-continuous group (G,° ,T) if conditions l) and 2) are satisfied, and (G, » ,T) will not be a topological group if and only if condition 3 ) holds. 1) (H,S) is locally compact and Hausdorff. 2) There is a group (K,•) and a homomorphism, h -> t^, from H into the automorphism group of K. 3) For some k in K, the set {h: th(x) = k] is not an open set in (H,S). 29 30 Corollary 3«3. If (H,•,S) is a locally compact Hausdorff group such that for at least one element x in H the set {h:hxh ^ = x} is not an open set, then (H,»,S) may be embedded in a left-continuous group which is not a topological group. Proof: The corollary follows from the theorem by letting (K,*) = (H,*) and t^(x) = hxh'*'1". The proof of the theorem will follow from a series of lemmas. In what follows, (H,»,S) is a topological group, (K,») is a group, and there is a map h -> t^ which is a homomorphism from H into the automorphism group of (K, •). Thus we have \ cV k)1 " V g 0 0 , th(kl k2 ) = t J W V * t (k) = k, ®H and th(eK) = for all h in H, and k in K, where e^ and e^ are the identity elements of (H,» ) and (K, •) respectively, and the group operation in the automorphism group of (K, •) is composition. It should also be noted that for each h in H, t. is one-to-one and onto. h Lemma 3.U. Let G = H X K = {(h,k): h € H, k 6 K}, and let an operation 31 be defined on G x G such that (hi,k1 )(h2 ,k2) = (^h^k-jt^kg)). Then G is a group with respect to this operation. Proof: Since ^ the defined operation is a binary operation on G. It is clear that (e^e^) is the identity in G, and that (h,k)- 1 = (h_1,t 1 (k“1 )). To show G is a group it remains to prove h that the binary operation is associative. Now (a,b)[(x,y)(w,z)] = (a,b)(xw,ytx(z)) = (axw,bta[ytx(z)]) = (axWjbtJyH^z)), and [(a,b)(x,y)](w,z) = (ax,bta(y))(w,z) = ( a x w ^ b tJyH^z)); hence the operation is associative. A topology T will be defined on G by defining a family of neighborhoods at each point, and showing that these neighborhoods form a base for a topology. Let neighborhoods of (e^,e^) be sets of the form H(eH,eK> ' Hh,eK>: h ^ V where N is an open neighborhood of e„ in (H,S). Let neighborhoods H H - of an arbitrary point (h0 A 0) in G be left translates of neighborhoods 32 of (e^,e^), that is, sets of the form H(ho,*o ) ■ = {(h h,k e^): h h € h N ] *■v o 3 o k ' o o etTJ II = C(h,kQ): h 6 h N }. Note that hQNe is an open neighborhood of hQ in (H,S). Hence the sets N^ X {kQ} are neighborhoods of (ho,kQ). Such sets form o a base for a topology (see Kelley [7]), since if 6 {(h,kQ): h^i^ } n {(h,kg): h 6 1^ ], then kx = kQ = kg, o 2 and h^ € N^ 0 N ^ = which is an open neighborhood of h^. Then {(h,^): h € N ^ } is an open neighborhood of (h^,k^) and C(h,kx ): h € Nh>} c {(h,kQ ): h € 1^ } n C(h,k2): h € 1^ }. i o 2 Let T be the topology induced by this base. Lemma 3«5« Left translations in (G,0 ,T) are homeomorphisms, hence continuous. Proof: The left translation (x,y) -» (a,b)(x,y) is a one-to-one onto mapping for each (a,b) and the topology is so defined that it is an 33 open mapping. The inverse map (x,y) •* (a,b) ^(x,y) = _jL(b*’^'))(x,y) is also an open map, hence each left translate a” and its inverse are continuous, and left translations are homeo morphisms of G onto G. Lemma 3.6. (G,T) is Hausdorff if and only if (H,£>) is Hausdorff. Proof; Assume (H,S) is Hausdorff. Then if (h^,k^) ^ (hg,kg) and k^ = kg, we have h^ r hg, and there are disjoint open neighborhoods and 0 g in S of h. and hg respectively. Then (h. ,k^) € 0^ X {kg}, (hg,kg) € Og x {kg} and x { ^ 1 fl 0g x {kg} = Lemma 3.7* (G,T) is locally compact if and only if (H,S) is locally compact. Proof: Assume (H,£>) is locally compact, and let (hQ,kQ) be an element of G. Then there is a compact neighborhood C of hQ in (H,S). It will be shown that C x {kQ} is a compact neighborhood of (hQ,kQ) in (G,T)."• If (0 QG ) is any open cover of C X {k O }, then (U CG ), where U = 0 fl (H X {k }), is also an open cover of C x {k }. Each U a a o o a is of the form {(h,kQ): h 6 ^ ? s} for some in S, and ( ^) is an open cover of C. Since C is compact, there is a finite subset, V ’V " ’Sx> which covers C, and the corresponding finite set '^l 2 n 0 ,0 ,*••.0 is a finite subset of the (0 ) which covers a, a ' ' a 'a,' 1 2 n C X {kQ}. Hence C X {kQ} is compact in (G,T), and (G,T) is locally compact. In a similar manner it can be shown that if (G,T) is locally compact, so is (H,S). Lemma 3.8. Right translations in (G, <> ,T) are continuous if and only if for every element k of K, the subgroup {h: h € H, = k} is an open set in (H,j3). Proof: Fix k € K. Consider the set o Pk - {h: th(kc ) = kD }. o The set P. is a subgroup of H, since if h_ ,h € P ., we have o 1 2 o * \ „ - l ^ o 1 = \ Cth -l(ko )] = \ < ko> “ V V a and lihj'1 6 P. . o Then for h* in H, h'P^ is a left coset of P^ , and o o h*Pk = (h'h: th(kQ) = k o} = Ch’h: thl(th(kQ )) = t ^ k , ) } = Ch: th(kQ) = th ,(ko)}. Now let 0 be an open set in (G, o ,T). We may assume 0 is a 35 base set of the form {(h,b): h € O' € £>} for some O' 6 S andsome b € K. Then the right translate of 0 by (hQ,ko) is the set 0(ho,kQ) = {(hho,bth(kQ)): hfo'es]. The set 0(hQ,ko) is open in (G,T) if and only if for each second component, bt^,(kQ ), that is, for each h ’ in O', the set Q* = {hhQ: bth(kQ ) = b t h ,(ko)} = {hh : t, (k ) = t. ,(k )} c o hv o h ,v o/J is an open set in (H,£a), and Q' is open just -when Q = Q'hQ ^ = {h: t^(ko ) = is an open set, since in the topological group(H, »,S), right translates of open sets are open. Now th(kQ) = th ,(kQ ) just when h € h'Pk , so Q = {h:th(kQ) = o t^Ck^)} = h'P^ , and Q is open just when P^ is an open set. o o Therefore, all right translates are continuous transformations just when Pk = {h: ^Ck) = k] is an open set in (H,S) for every k in K. Theorem 3*2 will now be restated and proved. Theorem 3*2. A topological group (H,»,S) may be embedded as a proper sub group in a left-continuous group (G, 0 ,T) if conditions l) and 2) are satisfied, and (G, ° ,T) will not be a topological group if and only if condition 3) holds. 1) (H,S) is locally compact and Hausdorff. 2) There is a group (K,*) and a homomorphism, h -» t^, from H into the automorphism group of (K,•). 3) For some k in K, the set {h: t^(k) = k} is not an open set in (H,S). Proof: Let (H,*,S) and (K,•) be a topological group and a group respectively that satisfy condition l) and 2), let (G, o ,t) be the product space H x K with group operation and topology as defined above in this chapter. Then, since (H,S) is locally compact and Hausdorff, by Lemmas 3*6 and 3»7j (G,T) is also locally compact and Hausdorff.By Lemma 3* 5j left translations in (G, o ,t) are continuous. Thus (G,® ,T) is a left-continuous group. By Lemma. 3.8, (G, 0 ,T) is a topological group just when condition 3) does not hold. The set H x {eK } = f(h,e^): h € H} is a subgroup of G, since (hl ,eK ^ h2,eiP = ^*h.h2 9 0K^h1^hg-l^ ®K ^ ^ = ,SK^ is in H X {e„}, and this subgroup is clearly isomorphic to (H, •) is. under the map h -» (h,e^.), and homeomorphic to H X lej.} tinder the subspace topology. Hence (H,•,S) is embedded in (G, o ,t), and the theorem is proved. 37 An example of a left-continuous group that is not a topological group and that has a topological group as a subgroup is given in Chapter V. CHAPTER IV A FIVE-R UNIQUENESS THEOREM During an investigation into the possibility that every left-continuous group satisfies Condition (I), an attempt was made to show that every left-continuous group with a metric topology satisfies Condition (i). Although neither a counter example to nor confirmation of this possibility was obtained, the investigation led to consideration of the Mickle-Rado 5-r condition for uniqueness [8], and a sharpening of the original Mickle-Rado uniqueness theorem was obtained. Several definitions will be needed for the development of the theory. In what follows, c(x,r) is the closed sphere of center x and radius r. Definition ^.1 A real-valued set function A, defined on all subsets of a separable metric space (X,d), is said to be a Caratheodory outer measure if i) for E c x , 0 < A(E) < ®, ii) A(6) = 0, iii) if E^ c Eg, then A ^ ) < A(Eg), iv) if E^,Eg,**«, is any sequence of subsets of X, 39 CO CO then A ( U E ) < Z A(E ), and n=l n n=l n v) if E^ and Eg are a positive distance apart, then A(E, U Eg) = A(E1 ) + A(Eg). Definition 4.2. Let (X,d) be a separable metric space, and let H be a family of homeomorphisms from X onto X. A Borel-regular Caratheodory outer measure A on X will be called H-invariant, or Haarwith respect to H, ifA[h(E)] =A(E) for everyE^ x and every h in H, and if there is a non-empty set 0 for which 0 < A(0) < <». Definition ^.3» Let (X,d) be a separable metric space, and let H be a family of homeomorphisms from X onto X. The family H is said to satisfy Condition (II) if for every pair of elements x,y in X there are homeomorphisms h^ and hg in H with h^(x) = y and hg(y) = x, and positive real numbers 4(x,y) and L(x,y) such that if 0 < r < 4, then 2 c(x,r) c hg[c(y,rL)] c c(x,rL ), and c(yjr) c h^[c(x,rL)] <= c(y,rL2). Definition U.U, A Caratheodory outer measure a on a separable metric space (X,d) is said to satisfy the strong 5-r condition if for every bounded set E there are positive real numbers k(E) and K(E) such that for every x in E if 0 < r < k(E), then c[c(x,5r)] < K(E) Definition U.5. A Caratheodory outer measure a on a separable metric space (X,d) is said to satisfy the 5-r condition at x if there are positive constants k(x) and K(x) such that for 0 < r < k(x), cr[c(x,5r)] < K(x)cr[c(x,r)]. If a satisfies the 5-r condition at every x in X, it is said to satisfy the 5-r condition on X, and it will be called a 5-r measure. Definition U.6. A Borel-regular Caratheodory outer measure A on X will be called locally finite if for each x in X there is an open set 0 such that x 6 0 and A(0 ) < °°. X X X If A is locally finite, then, since X is separable, there is a sequence of open sets • • • such that 00 (1 ) X = u On and A(On ) < «, n = l,2,.*«. n=l It is clear that if a measure satisfies the strong 5-r condition, it satisfies the 5-r condition. Note also that if A satisfies the 5-r condition at x, there is an open set containing x that has finite A-measure, hence if A satisfies the 5-r condition on X, then A is locally finite. The main uniqueness theorem which will be proved in this chapter can now be stated. 41 Theorem U.7. (Uniqueness theorem) Let (X,d) be a separable metric space, let H be a family of homeomorphisms from X onto X -which satisfies Condition (II), and let and Ag be Borel-regular Caratheodory outer measures on X that are Haar -with respect to H. Then if A^ satisfies the 5-r condition at one point in X, there is a positive real number c such that for every E c X, A2 (E) = cA^E). Theorem U.7 is a sharpening of the 5-r uniqueness theorem proved by Mickle and Rado [8] in 1959. Their theorem established the uniqueness of an invariant measure on the space of oriented lines in three-space, a result which does not follow from the standard ■uniqueness theorems for invariant measures. In their original theorem, one of the invariant measures is required to satisfy the strong 5-r condition. In the present theorem, this condition is weakened to the requirement that one of the invariant measures satisfy the 5-r condition at one point. First it will be shown that if a Haar measure satisfies the 5-r condition at one point, it satisfies the 5-r condition at every point in X, and then that if it satisfies the 5-r condition on X, it is unique. Theorem 4 . 8 . Let (X,d) be a separable metric space, let H be a family of homeomorphisms from X onto X satisfying Condition (il), and let A be an H-invar"iant outer measure on X that satisfies the 5-r condition at one point. Then A satisfies the 5-r condition everywhere on X. k2 Proof: Assume for 0 < r < k, A[c(xQ,5r)] < K A[c(xQ,r)], and let x be an arbitrary element in X. Then, since H satisfies Condition (il), there are positive constants &, v, L \, and h, h in ^xQ,x; ^x q ,x ; o in H with h(x)' = x^ o and h o (x) o = x such that if 0 < r < A, 2 c(x,r) c hQ [c(xo,rL)] c c(x,rL ) and c(xQ,r) c h[c(x,rL)] c c(xQ,rL2). Then for r < min (k,jJ), we have A[c(x,5r)] = A[h(c(x,5r))] < A[c(xQ,5rL)], and, letting n be the positive integer with 5n~^ < L < 5*1» A[c(xQ,5rL)] < A[c(xo,r5n+1)] < K2n+1A[c(xo,r5"n )] < K2n+1A[c(xo,rL_1) ] = K^ ^ A C h J c C x ^ r L " 1 ))] < K2n+1A[c(x,r)], and A satisfies the 5-r condition at an arbitrary x in X, hence on X. There are several different conditions that are sufficient for a Haar measure to satisfy the 5-r condition at a point, and they all concern an underlying Caratheodory outer measure which satisfies the 5-r condition, but which is not necessarily Haar itself. In what follows, this underlying measure will usually be denoted a, ^3 and use will be made of the upper and lower densities of a Haar measure A with respect to the underlying 5-r measure o. In Lemmas 4.9 through 4.13 below, (X,d) is a separable metric space, and a is a Caratheodory outer measure which satisfies the 5-r condition on X. Letting (2) = {x: x € X, 0 < r < l/n =* a[c(x,5r)] < ncr[c(x,r]]}, we have CO X = u A and A c A , n=l,2, * * *. , n n n+1 n=l Lemma 4.9 Let A be a Borel-regular Caratheodory outer measure on X, let t be a positive finite number, and let E be a subset of X. Set Gt(E) = {x:x € X.lim SUp > t}. Then for the sets A in (2), n cr[An n Gt(E)] < S A(E ). Proof; Since (3) An 0 Gt(E) c U Cc(x,r): x € An 0 Gt(E), r < A[c(x,r) 0 E] > tcr[c(x,r)]}, by a covering theorem of Mickle and Rado [9] * there exists a pairwise disjoint sequence of these spheres, 0(3^ , ^ ) , c(x2,r2 ),***, such that (k) 0 Gt(E) c U f c ( x i,5rj.): i=l,2,... Prom (2 ), (3), and (4) it follows that a[AXI 0 G.(E)] w < • 2- o[c(x. X ,5r. X )] *“< n • 2. cj[c(x.,r. X X )] 1=1 1=1 00 < I 2 A[c(xi,ri) n E] i=l 00 = £ At U c(x. ,r. ) n E] * i=l 1 1 < | ME). Lemma 4.10. Let A be a locally finite Borel-regular Caratheodory outer measure in X and let E be a A-measurable subset of X. Then for x 6 X - E, A[c(x,r) H E ] _ _ / _\ r^O " °M'x,r)T' = ° a,e‘ (a)- Proof; It suffices to show that (5) holds for A(E) < since letting En = E 0 0n, with 0n as in (l), we have t=c: X 6 X - z, limsvjp > 0) c t*: x € X - En, l i m ^ p «}, 45 and it suffices to show o(En ) = 0, n = 1,2 ,... Assume A(E) < °°. For 0 < t < «, let H.(E) = {x: x 6 X - E, lim sup —— yvE^ > t}. tw r _> o a[c(x,r)] For arbitrary e > 0, there is a closed set C such that C C E and A(E - C) < e. (See Morse and Randolph [10]). Since C is a closed subset of E, for r sufficiently small and for x 6 X - E, we have c(x,r) fl E = c(x,r) 0 (E - C), and for x € H^E), A[c(x,r) fl (E - C)] ^ j. lim sup - ■ r/~ \ i > b. r - 0 Hence H, (e ) c : G, (E - C) and by Lemma 4.9 for each positive integer n, t> tJ a[An n Ht(E)] < e H. Since e > 0 is arbitrary, for each positive integer n and t > 0, cr[A 0 H.(E)] = 0, and by (2), a[H, (e )] = 0. Then since n t u £x: x € X - E, limjmp > 0} = J J V » tE)' (5) follows, and the lemma is proved. Lemma 4.11. If A is a Borel-regular Caratheodory outer measure and E is a subset of X such that for x 6 E A E S S r f 1 - 116 then A(E) = 0. Proof: It suffices to show that A(A., fl E) = 0 for A as in (2). n n Since there is a sequence (0m ) of open sets such that 00 X = U 0 , tf(0 ) < °°, m = 1,2,..., m=l it suffices to show that A(A H O fl E) = 0. For x 6 E = n m nm A H O fl E, we have n m ' A[e(x,r) 0 Eng] , 0- r -» 0 o[c(x,r)] Let e > 0 be given. The family of closed spheres c(x,r/5) for which x € E ^ , r < l/n, c(x,r) cr o^, and A[c(x,r) 0 E^ ] < ecr[ c(x,r)] covers E ^ . Accordingly (by [93)j there is a pairwise disjoint sequence of closed spheres c(x^,r^/5)> cCx^r,-/;}), ... such that 0 9 CO E c U c(x. ,r.) and A(E ) = A[ U c(x. ,r.) H E ] nm . , x i ' nm . . ' x' x' nm x=l x=l 0 0 0 0 00 < 2 A[c(x.,r.) n E l < e 2 cr[c(x.,r )] < e n 2 a[c(x.,r./5)] ~ x=l 1 1 11111 i=l 1 1 i=l 1 1 = e n a[ U c(x ,r./5)3 < e n a(0). i=l Since e > 0 is arbitrary, n and m are fixed, and o(0m ) < », we have A(Em ) = 0. Lemma 4.12. Let A be a locally finite Borel-regular Caratheodory outer measure and let 1*7 E = [x: x € X, linjnjp = “). Then 0(E) = 0. Proof: It suffices to show c(E ) = 0, where E is defined as in N nm nm the proof of the previous lemma. Let t > 0 be given. By Lemma 1*.9> e(E ) < e[A n G, (0 )] < ? A(0 ), v nmy — n tv m t v m and since t is arbitrary, afE^) = 0. Lemma Let A be a locally finite Borel-regular Caratheodory outer measure on X and let E = {x: x € X, < ”> = ”1- Then A(E) = 0. Proof: By the previous lemma, wehave 0(E) =0. Let Ej = (x: x € E, lim inf < J}>j=l,2,... To show that A(E) = 0, it suffices to show A(E.) = 0, for j=l,2,..., J and to show that A(E.) = 0, it suffices to show that A(E. ) = 0, • d J ^ where E. = E. fl A. , with A asin (2). Let e > 0be given. There jn j n is an open set 0 such that E^n c 0 and a(0) < e. (See Mickle and Rado [9])* The set E. is covered by the family of closed spheres jn c(x,r/5) such that x 6 E. , r < l/n, c(x,r) c 0 , and A[c(x,r)] < jn jcr[c(x,r)]. Hence, by [9] there is a pairwise disjoint sequence of the spheres, c(x^,r^/5), c(x2 ,rg/5),•••» such that CO E C (J c(x. ,r. ). Then A(E.) < 2 A[c(x ,r )] < j 2 a[c(x ,r.)] Jn i=l 1 1 i=l < jn Z a[c(x. ,r./5)3 = jna[ U c(x. ,r /5)] i=l 1 i=l < jna(O) < jne. Since e > 0 is arbitrary, A(E^n ) = 0. It should be noted that Lemmas 4.9 through 4.13 are concerned only with the relationship between two measures on X, and that the family H of homeomorphisms is not involved. The following two lemmas concern properties of covers of X, and are also independent of the family H. Lemma 4 . l 4 . Let (X,d) be a separable metric space, A a locally finite Borel-regular Caratheodory outer measure on X, and cr a Borel-regular Caratheodory outer measure on X that satisfies the 5-r condition. If B is a Borel set with A(b) < a(B) > 0, and if S is a family of Bor el sets covering B with the property that for every x in B there is an S in S such that x € S and x — x t • A[c(x,r) 0 S i . _ lim sup v ' x J > 0 , r -» 0 a[c(x,r)] then there is an S in S with A(B 0 S) > 0. Proof; By Lemma 4.10, for a.e.(a) x in B, Mcferi n (X- E)J = r _ > 0 and since cr(B) > 0, there is such an x. For that x, its associated S = S, and any radius r, we have A[c(x,r) flS] ^ A[c(x,r) flS OB] , A[c(x,r) 0 s fl (X - B)] otcKr)] ^ ,V'B[efer)'I' "•---- oTofx/rti--- ^ A[c(x,r) fl S fl B] . A[c(x,r) fl (X - B)] 5 + rfcTx.OT" • Hence A[c(x,r) H S OB] ^ . A[c(x,r) O S ] ^ A lim sup — n — A > lim sup r 7 . r 0 cr[c(x,r)Jlvj/j ~ r -> 0 rf Lemma 4.1^. Let (X,d) he a separable metric space, 'A a locally-finite 50 Borel-regular Caratheodory outer measure, and a a Borel-regular Caratheodory outer measure that satisfies the 5-r condition. If B is a Borel set, and S is a family of Borel sets such that for each x in B, there is an S in S with x € S and ’ x — x A[c(x,r) fl Sx ] “ "Vo5 »t=(i,r)3 - >0’ then there is a sequence (S^), i=l,2,***, of sets in S with 00 Proof: \Je may assume a(B) > 0 , since if not the lemma is trivially true. First assume A(b) < “, and let (6) a = inf CA(B - US.): (S.) is a sequence of sets in S], i=l 1 • 1 Since A(B) < 00, a < ». Note that there is a sequence (S/*) with CO a = A(B - U S.*): take i=l 1 (S.*) = U {(S ): A(B - U S ) < a + l/n}. 1 n=l 1,n i=l 1,n Assume a(B - U S.*) > 0 . Then B - U S.* and the family i=l 1 i=l 1 £> satisfy the conditions for Lemma U.lU, and there is a set S in 00 00 S with A[(B - us.* ) n S] > 0, and A[B - (S U U S.*)] < a, i=l contradicting (6). 51 Hence a(B - IJ S.*) = 0. i=l 1 For arbitrary B, let Bn = B 0 0n, with 0n as in (l). For each 0 0 n, n=l,2,***, we have (S .), i=l,2,**», such that a(B - U S .) = 0. Ill H • ■ HI 1=1 Then a(B - U U S. )< o[ U (B - U S .)] < S a(B - U S ) = 0 . n=l i=d n=d i=l n^L i=a n In Lemmas ^.16 through U.2^ (X,d) is a separable metric space, H is a family of homeomorphisms from X onto X that satisfies Condition (il), A and a are Borel-regular Caratheodory outer measures that are Haar with respect to H, and cr satisfies the 5-r condition. Lemma ^-.16. Let E C X and let x € E such that A[c(x,r) fl E] ^ lim s u p ■ r t Vi ' > r -» 0 and let y € X. Then there is an h € H such that h(x) = y and > 0. 7.7 A[C(^ E)] Proof: Since H satisfies Condition (il), we have h € H, 4(x,y) and L(x,y) such that h(x) = y and if 0 < r < £(x,y), c(y,r) ch[c(x,rL)] p c c(y^^L )• We “ey assume L(x,y) > 1. Since a satisfies the 5-r condition, there is an n such that x 6 A and y € A , with A. as in ' n n n 52 (2). q-1 q Also there is a positive integer q such that 5 < L(x,y) < 5 • Note that a[c(x,rL) < a[c(x,5C^*)] < n^cr[c(x,5 < n^cr[c(x,r/L)] Then, for 0 < r < A(x,y)/nL, we have A[c(y,r) PI h(E)] A{h[c(x,r/L)] fl h(E)l _ A[c(x,r/L) 0 E] cr[c(y,r)] - alh[c(x,rL)]} cr[c(x,rL)J _ A[c(x,r/L) H E ] g[c(x,r/L)] a[c(x,r/L)] a[c(x,rL)J ^> -fcdS^Eyi—A[c(x,r/L) PIE] 1 ’ and A[c(y,r) n h(E)] ^ 1 . A[c(x,r) H E ] s n Ss?y.r)r ' > & atkx,£>r--> °- Lemma U.1T Let B be a Borel set in X such that A(b ) > 0 . Then there exists a sequence (h^), i=l,2, • • •, of homeomorphisms in H such that Proof: By Lemma If. 11, since A(B) > 0, there is a point xQ 6 B with liB SUp n *1 > 0 . r -> 0 cr[c(xo,r)] Then, by Lemma U.l6, for any x 5 X there is an h € H such that 53 h (x ) = x and x' o' A[c(x,r) n h (B)] lim s u p r 7--- pri > 0 . r 0 o[c(x,r)] Thus X is covered by the Borel sets h (b ) for x € X in a manner to satisfy Lemma 4.15, and there is a sequence (h^), i=l,2,***, in H such that a[X - U h.(B)] = 0 . i=l Lemma 4 . 1 8 . Let B be a Borel set in X such that a(B) > 0. Then there exists a sequence (iu ), i=l,2,..., of homeomorphisms in H such that 0 0 cr[X - Uh(B)]=0. i=l Proof; Since a satisfies all the conditions placed upon A, Lemma 4.17 holds with A - cr. Lemma jy a - Let B be a Borel set in X. Then c(B) = 0 if and only if A(B) = 0. Proof: Assume a is Haar with respect to H, hence cr(X) > 0. Thus if a(B) = 0, A(B) = 0. Assume A(B) = 0 and a(B) > 0. Then, by Lemma 4.l8, there is 00 a sequence (h.)# i=l,2,***, in H such that o[X - Uh.(B)] = 0, i=l 1 00 hence A[X - U^-(B)] = 0„ But since A is Haar with respect to i=l 1 H, A(X) > 0, and since ADnCB)] = A(B), we have A(b ) > 0, contra dicting the assumption. Hence if A(B) = 0, then a(B) = 0. The separable metric space (X,d), the sigma-algebra of Borel sets B, with either A or a restricted to the Borel sets, form sigma-finite measure spaces, and by Lemma ^.19# A is absolutely continuous with respect to ex. Hence the Radon-Nikodym Theorem may be applied (see Rudin [13])# and there is a Radon-Nikodym derivative, a real-valued Borel-measurable function f on Y, such that for every Borel set B A(B) = J f do. B In what follows, a Radon-Nikodym derivative, f, is fixed. Lemma ^-.20. For fixed t, 0 < t < °°, let L^ = {x: f(x) > t}, and let = {x: f(x) < t]. Then at most one of L^ and has positive a-measure. Proof; Assume a(l^) > 0. Then by Lemma ^.l8 there is a. sequence 55 00 (h.), i=l,2,***, in H such that a[X - Uh.(l», )] = 0o For each 1 • - 1 *0 1=1 i, i=l,2 ,***, if a[S^_ 0 h^(L^.)]> 0 , we have A[St n hi(l,t )] =J f do < to[St fl h^(L^)], and st D h.(Lt ) A[hi"1 (St ) fl Lt l = J f do > to[h1'1 (St ) fl lt], V 1 (st>n \ contradicting the invariance of A and o under H„ Hence, if a(L^) > 0, then cr(S^.) = 0. In a similar fashion it can be shown that if cf(S^) > 0 , then cr(L^.) = 0 . Lemma_Jk21. Let (7) c = sup £t: o£x: f(x) > t} > 0 }. Then 0 < c < ». Proof; If c = 0 , then A(x) = J f dcr = 0 , which contradicts the fact X that A is Haar with respect to H. Assume c = °°. Then for any Borel set B with A(B) > 0 , we have a(B) > 0, and by Lemma ^.20, for any t, o[B H {x: f(x)> t}] = a(B). Hence A(b) > ta(B) for any t; thus for every Borel set B, A(B) = But since A is Haar, there is a Borel set of positive 56 finite A-measure. Hence 0 < c < °°. Lemma_Jj^22o For a.e.(a) x in X, f(x) = c. Proof: The set 09 1 {x: f(x) > c] = U {x: f(x) > c + n=l and a{x: f(x) > c + = 0 , hence a(x: f(x) > c} = 0 . The set 00 [x: f(x) < c] = U {x: f(x) < c - n=l By the definition of c, a{x: f(x) > c - ^} > 0, for n=&,2,**», hence by Lemma if.20, a{x: f(x) < c - = 0 for every n, and a{x: f(x) ^ c) = 0 . Lemma if.23. For every Borel set B, A(B) = ca(B). Proof: A(B) = J f da = J c da = ca(B). B B Theorem if.2if. Let (X,d) be a separable metric space, let H be a family of homeomorphisms from X onto X which satisfies Condition (il), let 57 A and a be Borel-regular Caratheodory outer measures on X that are Haar with respect to H, and let a satisfy the 5-r condition. Then there is a positive real number c such that for every E c X, A(E) = ccr(E). Proof: Fix E c x . Then since A and a are Borel-regular, there are Borel sets B^ and Bg such that E c B^, E c Bg, A(E) = A(B^), and a(E) = c(Bg). Hence, with B = B^ H Bg, B is a Borel set, A(E) = A(B), and a(E) = a(B). By the previous lemma, with c defined as at (7), we have A(E) = A(B) = ca(B) = ccr(E). Theorem 4.7» (Uniqueness Theorem) Let (X,d) be a separable metric space, let H be a family of homeomorphisms from X onto X which satisfies Condtion (II), and let A^ and Ag be Borel-regular Caratheodory outer measures on X that are Haar with respect to H. Then if A^ satisfies the 5-r condition at one point in X, there is a positive real number c such that for every ECX, ^(E) = cA^(E). Proof: By Theorem 4.8, if A^ satisfies the 5-r condition at one point in X, then A^ satisfies the 5-r condition on X. Letting A^ be the a of Theorem 4.24, with c defined as at (7), we have Ag(E) = cA^CE) for every ECX. The density properties of a Haar measure A with respect to a 5-r measure a may be used to decompose the space X into disjoint subsets. Letting X = {x: lim inf = 0 }. ° r ^ 0 0c(x'r) X, = {x: 0 < lim inf A-P_(jl?-£ < lim sup Ac.(£;.r.). < os] ± r -> 0 0C(x 'r ) " r -* 0 0c(x>r) (8) X0 = {x: lim inf A9.(jE^.r.). < SUp A?.!*;1^). = and 2 r -> o 0c(Xjr) r -> 0 ac(x' r) X = [x: lim inf AP-l* ;.1*.). - oo] r -» 0 0c(x'r) we have X = xo uXx u X2 u x„ and the sets are pairwise disjoint. From Lemmas U.11, if.12, and ^.13* we have A(Xq U X^) = 0 and a(Xg U X j = 0 . If X^ = A(E) = A(E 0 A), and a(E) = a(E n B), where A = Xg U Xo and B = XQ if X1 = or B = XQ U X1 if is A-null, and similarly if X^^ is c-null. If this situation occurs, then both A and B are dense in X. This will be proved for the case X^ = <$; extension to the other cases is obvious. 59 Theorem b.25* Assume X = X U XL U X as in (8). Then X = clX = clX , o 2 09 o 00 where clE means the closure of E. Proof: Assume X ^ clX^. Then there is an x € X, and an open neighborhood 0 of x, such that 0 0 X = 6. Now X = U h(0 ), x x h € H x 00 and since X is separable, X = Uh . ( 0 ) for some sequence (h.), • a 1 X 1 1=1 00 i^L,2,***, of sets in H, and A(X) < L A[h.(0 )]. Since " i=l 1 X A[h.(0 )] = A(0 ) = A(0 H X ) = 0, we have A(X) = 0, which is 1 X X X 00 acontradiction, since A is Haar.Hence X = clX^. Similarly X = since a satisfies the 5-r condition, hence is positive on every open set. If the set X^ r ^>, then there is a point x in X at which the Haar measure A satisfies the 5-r condition, as is shown in the following theorem. Theorem b,26. Let (X,d) be a separable metric space, let a be a Borel- regular Caratheodory outer measure in X which satisfies the 5-r condition, let A be a locally-finite Borel-regular Caratheodory outer measure in X, and let x be a point in X such that 0 < lim inf < iim SUp Ajj-Sfe*.1!).! < «. r 0 a[c(x,r)J - r ^ a[c(x,r)J Then A satisfies the 5-r condition at x. 60 Proof: There are real numbers a and b such that « ^ „ A[c(x,r)] ^ . A[c(x,r)] ^ 0 < a = lim inf - ( • < iim sup — p = b < r -» 0 atc(x,r) J “ r ^ o a[c(x,r)J Fix e > 0 with e < a. Then there is an r such that for 0 < r < r o o A[c(x,r)] a[c(x,r)] Since a satisfies the 5-r condition, there are k and K such that r cr[c(x,5r)] < Kcr[c(x,r)] for 0 < r < k. Then for 0 < r < _2 , 5 r < k, we have A[c(x,5r)] < (b + e)cr[c(x,5r)] < (b + e)Kc[c(x,r)] < » * KACc(x*r)]> and A satisfies the 5~r condition at x. There is also a condition on a point in which ensures that A satisfy the p-r condition. Theorem 4.27. Let (X,d) be a separable metric space, let cf be a Borel- regulax Caratheodory outer measure in X which satisfies a ^-r condition, let A be a locally-finite Borel-regular Caratheodory outer measure in X, and let x be a point in X such that 2 ) there are positive constants rQ, m < 1 such that for 0 < r < rQ and for 0 < t < mr, A[c(x,t)] . A[c(x,r)] a[c(x,t)] - cr[c(x,r) J * Then A satisfies the 5-r condition at x. Proof; Fix x, where x satisfies l) and 2). Then x € An for some n , where An is defined as in (2 ), and for 0 < r < l/n, cr[c(x,5r)] < nc[c(x,r)]. For the m in 2), there is a positive integer t with 5”^ < m. Then for 0 < r < l/n, r < rQ /5> we have A[c(x,5r)] < A[cfx,rm)] < A[c(x,r)] < nt+1 A[c(x,r)] a[c(x,5r)] - a[c(x,rm)] - a[c(x>r5“t )] a[c(x,5r)] and A satisfies a 5-r condition at x. An additional condition imposed on the 5-r measure a will ensure that a Haar measure A satisfies the 5-r condition. Lemma 4.28. Let (X,d) be a separable metric space, let H be a family of homeomorphisms from X onto X which satisfies Condition (il), let A be a Borel-regular Caratheodory outer measure that is Haar with respect to H, and let a be a Borel-regular Caratheodory outer measure 62 that satisfies the 5-r condition and has the property that for x,y in X there are positive real numbers m and M such that for 0 < r < m, cr[c(x,r)] < Ma[c(y,r)], and tf[c(y,r)] < Mcr[c(x,r)]. Then there is a point x in X such that A satisfies the 5-r condition at x. Proof: If X1 4 A satisfies the 5-r condition on X. Assume X. = (|). Then X / (J) and X ^ d), by Theorem 4.25. «L ® O For x€X,y€X,we have m and M as above. Note that for co O 0 < r < m, |ja[c(x,r) ] < a[c(y,r) ] < Mcr[c(x,r)], and j|c[c(y,r)] < a[c(x,r)] < Mcr[c(y,r)]. Since a satisfies the 5-r condition, there is an n such that if 0 < r < l/n, a[c(y,5r)] < na[c(y,r)]. Since H satisfies Condition (il), there are £(x,y), L(x,y) and an h in H such that for 0 < r < £(x,y), we have h(y) = x and c(x,r) cr h[c(y,rL)]. There is a positive integer q such that 5^ ^ < L(x,y) < 5^» 63 Since lim inf g[ejy*r )j = there is a sequence (r^), i=l,2,* • •, lira r. = 0, r. _ < r., and lim ^.C.fo,rA.^. = 0. We may i -»°° 1 1 1 i crfcCy,^)] assume A[c(y,r^)3 < 1 for i = 1 ,2 ,* atcCy,^)] Let t > Mn^-. Then, since lim Apr-(& £ -H = “, there is an r ->0 aLC^x,r;j r such that for 0 < r < r , o o' A[c(x,r)] > t > Mn^. a[c(x,r)j Fix r with 0 < r < rQ, r < l/n, r < Ji(x,y), r < m, and cr[c(y,r;j Then A[c(x,r5”q-)] > tcr[c(x,r5-q-)] > to<10 [c(x,r5'"(1)] > Ma[c(x,r)] > cr[c(y,r)] > A[c(y,r)] = A{h[c(y,r)]} > A[c(x,r/L)] > A[c(x,r5"<1)] a contradiction. Hence, if X 4- The preceding lemmas are summarized in the following theorem. 6k Theorem k.2$. Let (X,d) be a separable metric space,let H be a family of homeomorphisms from X onto X -which satisfies Condition (il),let and Ag be Borel-regular Caratheodory outer measures on X that are Haar with respect to H, and let c be a Borel-regular Caratheodory outer measure on X which satisfies the 5-r condition,. Then if i), ii), or iii)holds, there is a positive number c such that for every E cr x, A^(E) = cAg(E). i) There is a point x in X such that 0 < lim inf < lim sup < ®. r 0 alc(x,r)] “ r .jo e[c(x,r) j ii) There is a point x in X such that Al C v A JL' I I lim inf —>■ r vf = w, and there are positive constants r -> 0 aLc'v r and m < 1 such that for 0 < r < r and 0 < t < rar, o o * A[cfx,t)3 A[c(x,r)] a[c(x,t)] - a[c(x,rj] * iii) For every pair of points x,y in X there are positive real numbers m and M such that for 0 < r < m, cr[c(x,r)] < Mcr[c(y,r)], and a[c(y,r)] < Mcr[c(x,r)]. CHAPTER V AN EXAMPLE In this chapter is an example which illustrates several of the theorems presented in the previous chapters, and demonstrates that left-continuous groups that are not topological groups do indeed exist. Following the example is a theorem on the decomposition of a measure into the sum of two measures, one trivially infinite or zero, the other unique, and infinite only on sets with subsets of finite positive measure. Example 5.1. Let G = {(x,y): x,y € R , x > 0}. For (a,b),(x,y) 6 G, let (a,b)(x,y) = (ax,ay+b). Clearly (G, o ) is a group, with (1,0) the identity, and for each (x,y) in G, (x,y) -1 = (l/x,-y/x). If G is mapped onto the set of all 2 x 2 matrices of the form then the group operation corresponds to matrix multiplication, and left translates of horizontal lines are horizontal lines, but right translates of horizontal lines are lines which are not necessarily horizontal. Let the topology T on G be the topology with neighborhood base 66 at (x q ,y q) consisting of sets of the form f(x,yo): x 6 (a,b), 0 < a < < b < »). Open sets are then unions of open intervals on horizontal lines. With the group operation and topology as defined above, (G, o ,t ) is a left-continuous group which is not a topological group. To show that left translations are continuous, it suffices to show left translates of open base sets are open base sets, since for any open set 0, (x,y)~3'0 = (l/x,-y/x)0. Then, since (t,w){(x,yQ): x 6 (a,b)} = {(tx,tyQ + w): x S (a,b)} = {(x^): x 6 (ta,tb), y1 = tyQ + w}, left translates of open base sets are open, and left translations are continuous. It is clear that (G.T) is locally compact and Hausdorff, so (G,o ,t) is a left-continuous group. To show that (G, ° ,T) is not a topological group, we need to show that not all right translations are continuous. Consider the open set 0 = {(x,l): l < x < 3 } and right translation by (1 ,2 ). Then 0(1,2) = C(x,2x+l): 1 < x < 3) is an open segment of the line y = 2x+l, of slope 2, and contains no open sets. Hence right translations are not all continuous and (G, ° ,T) is not a topological group. The left-continuous group (G,° ,T) satisfies Condition (i), since if 0 is an open set containing (1,0), the identity in G, there is a set U c 0, U = {(x,0): a < x < b) for some a and b with 0 ‘ E ™ have V' 1 = t(x,0): < jij}, and W ' 1 - C(x,0): c U c 0. Since (G, ® ,T) satisfies Condition (i), there is a left- translation-invariant Borel-regular outer measure on G that is positive on open sets and finite on compact sets. For open base sets, let A[{(x,yQ): a < x < b)] = In |, A((j») = 0, where In is the natural logarithm, and for arbitrary E c g , letting B^, i=l,2 , •••, be an open base set, let CO CO A(E) = inf{ £ A(B.): E c (J B.} if there is a sequence of open i=l 1 i=l base sets that covers E, and A(e) = 09 otherwise. Clearly A is aBorel-regular outermeasure that is positive on non-empty open sets and finite on compact sets, and it is invariant under left translation, since ■ A[(t,w)C(x,yQ): a < x < b}] = AtfCx^): y1=tyQ + b, ta < x < tb}] = In || = In | = A{(x,yQ): a < x < b}. This example serves as an illustration of the embedding theorem. Let (H, .,S) be the group of positive real numbers under multiplication, with the subspace topology induced by the usual topology on the reals. Then (H, »,s) is a locally compact Hausdorff topological group. Let (K,+) be the group of real numbers under addition. Then G = HXK. . For h € H, define the map h -> t^ by t (k) = hk. For each h, t^ is 68 an automorphism of K, since t^(x+y) = h(x+y) = hx+hy = t^(x)+t^(y), and t^ is a one-to-one onto map. If t^(x) = t^(y), then hx=hy, x=y, and t^ is one-to-one. For y 6 K, y / 0, we have y = t^(y/h), and for y = 0, y = t^Co) for any h. Since th ^ (x) = = tfa (hg(x)) = t^ [t^ (x)], the map h -> t^ is a homomorphism from H into the automorphism group of K. The group operation on G, (a,b)(x,y) = (ax,ay+b) = (ax,b+t (y)), is as defined in the proof of the embedding theorem in Chapter III. It is clear that for k / 0, the set {h: ^ 0 0 = k} = {l] is not an open set in H, so the embedding left-continuous group is not a topological group. Now the set {(x,0): x € H) is a subgroup of G which is isomorphic to H, and the induced subspace topology is homeomorphic to the topology on H, both under the obvious map x -*■ (x,0), hence (H,»,S) is embedded in (G, °,T). One non-trivial left-translation-invariant outer measure, A, has been defined on G. Another non-trivial left-translation- invariant outer measure also exists: for E C G , let cr(E) = sup{A(F): F c E, A(F) < “}. Then a is clearly an outer measure on G, since cr((j>) =0 , a is CO monotone, and for any sequence of sets E in G, and any F c (J e n n=l n 00 00 with A(F) < °°, we have A(F) = A(F HUE) = A( IJF 0 E ) < n=l n n=l n 00 00 CO E A(F fl E ) < E a(E ), and since Fwas arbitrary, cr( U En) < n=l n n=l n n=l 69 CO E a(E ), and a is countably sub-additive. It is clear from the n=l n definition that cr is left-translation-invariant. There are sets on which A is infinite and cr is not; for instance, any vertical line segment has infinite A-measure and zero a-measure. The relationship between measures such as A and cr above has been discussed by Berberian [l], and a theorem concerning this relationship is given below. In what follows, (X,M,A) represents a measure space, where M i s a sigma-algebra of subsets of X and A is a measure on M. Definition 5.2. A set S in M is said to have the finite-subset property if there is a set A in M with A C S and 0 < A(A) < “. Definition 5«3» A set S in M is said to be locally-null if for every set F in M with A(F) < “, A(S fl F) = 0. Theorem !?.*<•. Let (X,M,A) be a measure space. Then there are measures A and Am00 on M M such that i) A = Ac + A ^ ii) for E € M, AjE) = 0 or AjE) = «, iii) for E? M, ^C (E) = 00 if andonly if sup{A(F): F € M, F c E., A(F) < » ) = »,and 70 iv) Aq is unique. Proof: Let A and A "be defined on M as follows: For E € M, let C 09 — — Ac (e ) = sup{A(F): F ?H, FCE, A(F) < »}, f » if E contaii•tins a non-null locally-null set, A09' .<»> ’ " ] I 0 otherwise. Clearly (($) = 0 and Aw((J) = 0, so to show that Aq and A^ are measures it suffices to show that they are countably additive. Note that A (E) < A(E) for E € M, and that if A(E) < », then A(E) = A (E), C __l " c and AjE) = 0. Let (E ), n=l,2,•••, he a sequence of pairwise disjoint sets ** 00 CO in M. First assume A(E ) < » for all n. Then £ A (E ) = E A(E ). — n' n=x , c' n' n=x , n' 00 CO 00 CO 00 If E A(E ) < “, ire have E A (E ) = E A(E ) =A(UE) =A(UE). n=l n n=l c n n=l n n=l n c n=l n 08 m m m If E A(E ) = », then for all m, EA(E ) = E A (E ) = A ( U E ) < ~, n=l n n=l n n=l c c n=l n m m <» 09 andlim A ( U E ) = lim EA(E ) = A( U E ) = «. Since A ( U E ) m-*» c n=l n m-*» n=l n n=l n c n=l n m co oo > A ( U E ) for all m, we have A ( U E ) = £A(E), for the case ~ c n=l n c n=l n n=l c n where Ate ) < » for all n. n «00 Next assume for some n, A(E ) = ». Let F c U E , F € M. n v.n=JL 1 “ — ^ A(F) < ». Then A(F D E^) < » for all n, and 00 09 A(F) = S A(F fl E ) < E A t e ) , n=l n n=l 00 CO so, since F is arbitrary, A ( U E ) < E A (E ). Let F c: E , F € M, n=l ~ ~ n=l c n n n' n 7 1 m m A(F ) < 03. Then 2 A(F ) = A(UF) 09 09 09 00 S A (E ) < A ( U E ), and E A (Ej = A ( U e J. Hence A„ is countably n=lc " ' c n=l" n=l c ° . 4 " c additive, and is a measure on M. It is clear from the definition of A that every set in M that has positive A -measure has the finite- C T Q subset property. No\r A is countably additive, since, letting (E ) be a sequence CO of pairwise disjoint sets in M, we have, first, if A ( U E ) = 0, 09 - n n=i 09 there is no non-null set in U E , hence none in E for any n, n=l n n CO hence A (E ) = 0 for every n, and 2 A (E ) = 0. Second, if os' n 7 „ , “ n 7 n=i CO CO A ( U E ) = », there is a locally-null non-null set L cr U E . Then “ n=ivs-1 n n=l V, 1 n I (1 E is locally-null for each n. If L fl E is null for each n, n n 09 09 then 0 = 2 A(L nE)=A(ULflE) = A(L), contradicting the fact n=l n n=l n that L is non-null. Hence for some n, 1 fl E is locally-null and CO ** non-null, and 2 A (E ) = ». Hence A is countably additive and is n=l 09 n “ a measure. For every E € M, A(E) = A (E) + A (E), since first, if A(E) < os, c ® we have Ac (E) = A(E) and Ara(E) = 0. Second, let A(E) = os. if Aw(E) = os, then clearly A(E) = Ac (E) + A^(e ) . So assume Ara(E) = 0 and Ac (E) = R < o. If R = 0, then E is locally -null and non-null, and 72 A (E) = ea, a contradiction. For R > 0, there is a sequence of sets m F € M, F e E, A(F ) > R - l/n. Then A( U F ) > R - l/m for all m. n — 7 n 7 ' n 7 7 ' - n ' 7 n=i m m eo Not/ A( U F ) < A (E) for all m, hence lim A( U F ) = A( U F ) < n=l n c at** n=l n n=l n CO 09 Ac(E). Then A(E - U ?n) = If for some Fq c E - U Fq , 0 < A(Fq) CO < then A (E) > A(F ) + A( U F ) > R, contradicting A (E) = R. c ° n=l a CO So (E - U F ) is a locally-null non-null subset of E, and A (E) = », n=l n contradicting A (E) = 0. Therefore ire have A.(E) = », and it has co C been shown that A = AC + Aoo . It remains to show that A is unique. Assume h + A_ = A = c c 00 a + cr . where A and o satisfy iii). and A and cr satisfy ii) c co c c M 00 00 in the statement of the theorem. Assume A / a . Then there is a set E If A (E) = ", then A (E) = A(F ), where F = U {F : A.(E) < oo c o o n c A(Fn) + l/n,Fn € M, Fn c E}, andAw(Fo) = 0, a j F Q) = 0, and a (F ) = A (F ). Since A (E - F ) = «, and A (E - F ) = 0, E - F c o c' O7 wv o7 C O o is locally-null, non-null, hence cr (E - F ) = 0, and a_(E) = A (E), C O c c contradicting the assumption that A / c . c c The measure A^ is not necessarily unique; consider the following case. Let (X,M,A) be a non-sigma-finite measure space, assume A = A , ' v 73 and define A, as follows: A^(E) = “ if E is not sigma-finite, and A^(e ) = 0 if E is sigma-finite. Then is clearly a measure, which assumes only the values 0 and «, and A = + A-j_. But also if Ag(E) = 0 for all E € M, then A = Aq + A^, and Ag ^ A^. It is of course only in non-sigma-finite spaces that Theorem 5 A has any application. * The following questions have arisen in the course of the research for this dissertation: 1) Does every left-continuous group satisfy Condition (l)? 2) Does every left-continuous group have an open subgroup which is a topological group? 3) If a left-continuous group satisfies Condition (i), does it then have an open subgroup that is a topological group? h) Does there exist a separable metric space endowed with two Borel-regular Caratheodory outer measures, A and cr, where A is locally-finite, and a satisfies the 5-r condition, and for no point x in X i3 < « ? BIBLIOGRAPHY 1. S. K. Berberian. "Counterexamples in Haar Measure." American Mathematical Monthly, 7 3, no. U, Part II. (April, 1966)f 135-%. 2. ______. Measure and Integration. New York. Macmillan. 196 5. 3. N. Bourbaki. General Topology, Part 1. Reading, Massachusetts. Addi son- Wesley.' 19^6. 4. R. Ellis. 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