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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 . The
intersection ...
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 . The
intersection ...
Page 166
If we put E ( E ) = { FeE | 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 & is a o - field , then E ( E
) is a o - field . Σ ( Ε ) is called the restriction of Σ to E . If Σ , is a field , Εε Σ . , and ...
If we put E ( E ) = { FeE | 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 & is a o - field , then E ( E
) is a o - field . Σ ( Ε ) is called the restriction of Σ to E . If Σ , is a field , Εε Σ . , and ...
Page 201
There should be no confusion between the symbols & and Eq , for if a reduces to
the single elementa then the field E , is the o - field Eq . An elementary set in S is
one of the form Paca Ec where E , is in and , for all but a finite number of a , Ec ...
There should be no confusion between the symbols & and Eq , for if a reduces to
the single elementa then the field E , is the o - field Eq . An elementary set in S is
one of the form Paca Ec where E , is in and , for all but a finite number of a , Ec ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero