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Page 106
The functions totally u - measurable on S , or , if u is understood , totally measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in { with v ( u , E ) < oo , the product Xet ...
The functions totally u - measurable on S , or , if u is understood , totally measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in { with v ( u , E ) < oo , the product Xet ...
Page 119
Then we say that f is u - measurable if the function g defined by g ( s ) = f ( 8 ) if 8 ¢ N , g ( 8 ) = 0 if s eN , is u - measurable . A discussion similar to that preceding Definition 6 shows that by considering this somewhat more ...
Then we say that f is u - measurable if the function g defined by g ( s ) = f ( 8 ) if 8 ¢ N , g ( 8 ) = 0 if s eN , is u - measurable . A discussion similar to that preceding Definition 6 shows that by considering this somewhat more ...
Page 178
Thus we may and shall assume that v ( u , F ) < oo and v ( 2 , F ) < oo . Since g is 2 - measurable there is a sequence { n } of simple functions converging to g ( s ) for every s in F except on a set ECF with v ( 1 , E ) = 0 ( by ...
Thus we may and shall assume that v ( u , F ) < oo and v ( 2 , F ) < oo . Since g is 2 - measurable there is a sequence { n } of simple functions converging to g ( s ) for every s in F except on a set ECF with v ( 1 , E ) = 0 ( by ...
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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