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Page 98
The total variation of an additive set function defined on a field of subsets of a set S is also additive on E. PROOF . Let { A ; } be a finite sequence of disjoint sets in E with ACEUF where E , F € ? and EF = $ .
The total variation of an additive set function defined on a field of subsets of a set S is also additive on E. PROOF . Let { A ; } be a finite sequence of disjoint sets in E with ACEUF where E , F € ? and EF = $ .
Page 101
SES SES If for some null set N the restriction off to S - N has its values in a separable set , the function f is said to be jl - essentially separably valued . Expressions such as essentially compact valued are defined similarly .
SES SES If for some null set N the restriction off to S - N has its values in a separable set , the function f is said to be jl - essentially separably valued . Expressions such as essentially compact valued are defined similarly .
Page 182
Q.E.D. Note : The unique u - integrable function of Theorems 2 and 7 is called the Radon - Nikodým derivative of 2 with respect to u and is often denoted by da / du . Thus da / du is defined u - almost everywhere by the formula ( ( da 2 ...
Q.E.D. Note : The unique u - integrable function of Theorems 2 and 7 is called the Radon - Nikodým derivative of 2 with respect to u and is often denoted by da / du . Thus da / du is defined u - almost everywhere by the formula ( ( da 2 ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
quences | 26 |
Copyright | |
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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero