THE THEORY OF INNER MEASURES: AN AXIOMATIC APPROACH by John C. N. Chan Mathematics Report #2-76 ProQuest Number: 10611591 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. ProQuest ProQuest 10611591 Published by ProQuest LLC (2017). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346 Copyright John C.N. Chan 1976 THE THEORY OF INNER MEASURES: AN AXIOMATIC APPROACH A thesis submitted to Lakehead University in partial fulfillment of the requirements for the Degree of Master of Science JOHN C. N. CHAN 1975 ACKNOWLEDGEMENT I wish to thank my supervisor. Professor W. Eames, for his advice, encouragement and patience during the prepartion of this thesis. 1 TABLE OF CONTENTS Page ABSTRACT ii NOTATION AND TERMINOLOGY iii INTRODUCTION: SOME HISTORICAL REMARKS I CHAPTER I: DEFINITION OF INNER MEASURES 4 CHAPTER II: CHARACTERIZATION OF INNER MEASURABILITY AND INDUCTION OF INNER MEASURES 19 CHAPTER III: CONTRACTION OF INNER MEASURES AND INDUCTION OF OUTER MEASURES 29 CHAPTER IV: SEQUENCES OF CONTRACTIONS OF INNER MEASURES 39 REFERENCES 43 ABSTRACT This thesis presents an axiomatic approach to the theory of in- ner measures. In Chapter I, we recognize that an axiomatic approach analogous to that for the theory of outer measures is not appropriate: such an approach involves only a finiteness concept, and consequently, as soon as countable collections are involved, it fails. It is then natural for us to speculate that our model should permit the change of limits. This idea leads us to the definition of an inner measure. In Chapter II, we consider a space of finite measure and charact- erize inner measurability. We also prove that the definitions of inner measurability given by Young and Caratheodory are equivalent, and that Lebesgue’s definition of measurability is equivalent to those given by Young and Caratheodory. Then, the assumption that the space has finite measure is dropped, and we study the inner measures induced by a measure. In Chapter III, inner measures are contracted so as to guaran- tee that they are countably additive over their classes of inner measurable sets, and so that they always generate an outer measure. The last part of this chapter deals with conditions under which the set function generated by an inner measure y^ is a regular outer measure. Finally, in Chapter IV, some relation between a sequence of contracted inner measures and the associated measure spaces is estab- lished . NOTATION AND TERMINOLOGY Throughout, X is given space. P(X) is the power set of X. R = [_oo^oo], A set function is a function from a subclass of P(X) to R. If there can be no ambiguity, the intersection sign is omitted. A* is the complement of A. LEBESGUE'S CONDITION FOR MEASURABILITY: p. 2. YOUNG’S DEFINITION OF INNER MEASURABILITY: p. 2. OUTER MEASURE: Definition, p. 2. SUPERADDITIVE SET FUNCTION: Definition, p. 4. ALGEBRA: A subclass of P[X) which is closed under finite unions and complementation, p. 4. o-ALGEBRA: An algebra closed under countable unions, p. 4. p. 4. : A set of all positive integers, p. 4. |A|: Number of elements in the set A, p. 4. MONOTONE INCREASING: A set function y is monotone increasing (or simply called ’monotone’) if whenever A => B, y (A) ^ y (B) , p. 4 COUNTABLY SUPERADDITIVE: A set function y is countably superaddi- tive over P(X) if, whenever {B } is a sequence of pairwise oo oo disjoint sets from P(X), y( u B ) > y(B ), p, 5. n=l n=l n: p. 5. n-MEASURABLE: Definition, p. 5. CARATHEODORY’S METHOD OF DEFINING MEASURABILITY: p. 5. M(ri): p. 6. iii ADDITIVE OVER A CLASS OF SETS C; Definition, p. 7. FINITELY ADDITIVE: A set function y is finitely additive over a subclass of P(X) if, whenever A^^,... ,A^ are pairwise disjoint sets from the subclass whose union is also in the ri n subclass u A.) = I y(A.)s p. 9. i=l ^ i=l ^ COUNTABLY ADDITIVE: A set function y is countably additiver over a subclass of P(X) if, whenever is a sequence of pairwise disjoint sets from the subclass whose Union is also in oo c» the Subclass, ( u B ) = y(B ), p. 9. n=l ^ n=l rj-NULL: Definition, p, 10. U-NULL: p. 10. : p. 10 . ]^-MEASURABLE: p. 10. y^ = INNER MEASURE: Definition, p. 11. A DECREASING SEQUENCE OF SETS: is a decreasing sequence of sets if Bj^ => B2 => . ., p. 11. i/m CONDITION: p. 11. MONOTONE AND BOUNDED : A real sequence ^ is monotone if - ^n ^ (called monotone increasing), or s for all n (called monotone decreasing). is bounded if there exists a number K such that lx I < K ’ n' for all n, p. 11. CARATHE0b0RY*S INNER MEASURE: Definition, p. 12. iv H*: p. 12. y*: p. 12. a,3: p. 12. a-MEASURABLE KERNEL: Definition, p. 12. a-MEASURABLE, 3-MEASURABLE: p. 12. U*-MEASURABLE KERNEL: p. 12. y*-MEASURABLE: p. 12. M(y*) : p. 13 . y*-MEASURABLE: p. 13. lim y*(K ): The limit superior of a sequence of real numbers defined n-x» as lim y*(K ) = inf sup y*(K ), p. 13. n-x» ^ n>k ^ yi = LEBESGUE'S INNER MEASURE: Definition, p. 14. INCREASING SEQUENCE OF SETS: increasing sequence of sets if B]^ c B2 c ..., p. 16. Cy^): p. 16. y-MEASURABLE KERNEL: p. 19. a-MEASURABLE COVER: Definition, p. 19. y*-MEASURABLE COVER: p. 19. REGULAR OUTER MEASURE: Definition, p. 20. M(y*): p. 25. X y : Definition, p. 26. yg: Definition, p. 26. y : p. 26. MEASURE: A measure is a countably additive, non negative, set function, p. 27. V RING: A subclass of sets of P(X) closed under union and differ- ence, p. 27. p. 27. M(yo): p. 27. restricted to M(y^) (used systematically throughout), p. 27. dol^CUo): P- 27. y : Definition, p. 27. yo-MEASURABLE KERNEL: p. 27. y-MEASURABLE KERNEL: p. 27. y^ = A CONTRACTION OF y^: Definition, p. 29. y^-MEASURABLE: Definition, p. 29. M(y^): p. 30. C y : Definition, p. 31. ycl^CPc^' P- y^: p. 34. M(y^): p. 34. y^|M(y^): p. 34. y^-MEASURABLE KERNEL: p. 35. y°: Definition, p. 35. y^-MEASURABLE COVER: p. 36. 00 {C^}: A sequence of y^-measurable sets such that u is i=l finite, p. 39. {y }: A sequence of inner measures associated with {C.}, p. 39. vj • 1 1 (M(y^ ^ measure space (a a-algebra ) together with a v^i measure y^ on it), p, 39. '^i vi CO M(y [Here, uC. is the shorthand for u C.) p. 39. ^ i=i ^ p. 39. p. 39. y^-NULL: p. 42. Vll INTRODUCTION: SOME HISTORICAL REMARKS In 1898, Emile Borel [1] gave a descriptive definition of a measure as follows: (1) It 'is a funat-ion from a otass D of subsets of the real line to [0^°^); D is closed under differences and finite unions; (2) the measure of the union of a finite number of pairwise disjoint sets from D equals the sum of their measures; (3) the measure of the difference of two sets from fD (a set and a subset) is equal to the difference of their measures; (4) every set whose measure is not zero is uncountable, The existence of such a descriptive measure can be seen by tak- ing iV to be the class of all finite unions of intervals from the real line, taking the length of each interval to be its measure and extending this in the obvious way. However, the description did not include the idea of countable additivity. Based upon Borel’s ideas, H. Lebesgue in 1901 [5] defined a measure for any set in the interval [a,b] to be a non-negative number satisfying the follow- ing conditions: (1) Two congruent sets have the same measure; (2) the measure of the union of a finite or countable number of pairwise disjoint sets is the sum of the measures of the summands; (3) the measure of the set [0^1] is 1. The importance of countable additivity for a measure was 1 2 realized; however, the three conditions are incompatible as there do exist non-measurable sets in [0,1]. (Lebesgue did not know this at the time: the first non-measurable set was constructed by Van Vleck in 1908 [15].) Taking the length of a segment [a,b] or of the interval to be its measure m, Lebesgue gave a con- structive definition of a measurable set. He first defined the outer measure of each set A <= [a,b] by m (A) = inf{\m((a.jh.)): A c \j(a .^b.)}. He then defined the inner measure m^[A) of A to be b-a-m^([a,b]-A) and defined A to be measurable if m^(A) = m^(A). Lebesgue's definitions for outer and inner measures were also given by G. Vitali in 1904 [14], and by W. H. Young, also in 1904 [17]. However, Young defined measurable solely in terms of an outer measure m : e a set A ts sa-id to be outer-measurable if and only if, for aVl D such that Ar\D = 0, m^(A)-Hn^(D) = m^(AuD).