Linear Operators: General theory |
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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
If we put E ( E ) = { F € 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 € X , and that if Eis a o - field , then E ( E ) is a o - field .
If we put E ( E ) = { F € 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 € X , 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 Ey . The family of all sets in S of the form S , XER with Ε ε Σ will be denoted by Σπ . 18 LEMMA . For each a the family In is a field of sets in S and Σ . = U ...
The symbol will be used for the o - field of sets in S determined by Ey . The family of all sets in S of the form S , XER with Ε ε Σ will be denoted by Σπ . 18 LEMMA . For each a the family In is a field of sets in S and Σ . = U ...
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Contents
Preliminary Concepts | 1 |
The VitaliHahnSaks Theorem and Spaces of Measures | 7 |
B Topological Preliminaries | 10 |
Copyright | |
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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed closure complex condition Consequently contains continuous functions converges Corollary defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function f given Hence Hilbert space implies integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space linear topological space Math means measure space metric space neighborhood norm open set operator problem Proc projection Proof properties proved range reflexive respect Russian satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero