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Page 136
There is a uniquely determined smallest field and a uniquely determined smallest o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all subsets of S , which contains a given family t ...
There is a uniquely determined smallest field and a uniquely determined smallest o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all subsets of S , which contains a given family t ...
Page 166
Suppose that E is a set in E. If we put E ( E ) = { F e E | F C E } it is clear that E ( E ) is a field of subsets of E , and that E ( E ) is the family of all sets AE , A € E , and that if Eis a o - field , then E ( E ) is a o - field ...
Suppose that E is a set in E. If we put E ( E ) = { F e E | F C E } it is clear that E ( E ) is a field of subsets of E , and that E ( E ) is the family of all sets AE , A € E , and that if Eis a o - field , then E ( E ) is a o - field ...
Page 201
The symbol will be used for the o - field of sets in S determined by 7. The family of all sets in S of the form Sq , XER with E , ε Σ will be denoted by Σπ . Q € 18 LEMMA . For each r the family en is a field of sets in S and Σ , = Ο Σπ ...
The symbol will be used for the o - field of sets in S determined by 7. The family of all sets in S of the form Sq , XER with E , ε Σ will be denoted by Σπ . Q € 18 LEMMA . For each r the family en is a field of sets in S and Σ , = Ο Σπ ...
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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