Linear Operators, Part 1 |
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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 ) < 00 , the product met of with the
characteristic ...
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 ) < 00 , the product met of with the
characteristic ...
Page 119
( b ) the function g defined by g ( s ) = f ( s ) if s & S + US - , g ( s ) = 0 if se S + US -
, is u - measurable . Next suppose that we consider a function † ( vector or
extended real - valued ) which is defined only on the complement of a u - null set
NCS .
( b ) the function g defined by g ( s ) = f ( s ) if s & S + US - , g ( s ) = 0 if se S + US -
, is u - measurable . Next suppose that we consider a function † ( vector or
extended real - valued ) which is defined only on the complement of a u - null set
NCS .
Page 178
Thus we may and shall assume that v ( u , F ) < oo and v ( 2 , F ) < 0 . Since g is a
- measurable there is a sequence { 8n } of simple functions converging to g ( s )
for every s in F except on a set E ÇF with v ( 2 , E ) = 0 ( by Corollary 6 . 13 ( a ) ) .
Thus we may and shall assume that v ( u , F ) < oo and v ( 2 , F ) < 0 . Since g is a
- measurable there is a sequence { 8n } of simple functions converging to g ( s )
for every s in F except on a set E ÇF with v ( 2 , E ) = 0 ( by Corollary 6 . 13 ( a ) ) .
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |
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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint 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 obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero