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Page 193
JS IJT 12 LEMMA . Let ( R , ER , Q ) be the product of finite measure spaces ( S ,
E , u ) and ( T , ET , 2 ) . Let E be a g - null set in R . Then for 2 - almost all t , the
set E ( t ) = { s ( s , t ) € E } is a u - null set . Proof . By Lemma 11 it may be
assumed ...
JS IJT 12 LEMMA . Let ( R , ER , Q ) be the product of finite measure spaces ( S ,
E , u ) and ( T , ET , 2 ) . Let E be a g - null set in R . Then for 2 - almost all t , the
set E ( t ) = { s ( s , t ) € E } is a u - null set . Proof . By Lemma 11 it may be
assumed ...
Page 410
1 Definition. A set K Q X is convex ifx,ye K, and 0 ^ a ^ 1, imply ax-\-(l — a)y e K.
The following lemma is an obvious consequence of Definition 1. 2 Lemma. The
intersection of an arbitrary family of convex subsets of the linear space X is
convex.
1 Definition. A set K Q X is convex ifx,ye K, and 0 ^ a ^ 1, imply ax-\-(l — a)y e K.
The following lemma is an obvious consequence of Definition 1. 2 Lemma. The
intersection of an arbitrary family of convex subsets of the linear space X is
convex.
Page 697
Nelson Dunford, Jacob T. Schwartz. 11 LEMMA . Let ( S , E , u ) be a positive
measure space and let { T ( 4 , . . . , tk ) , t , . . . , tx > 0 } be a strongly measurable
semi - group of operators in L ( S , E , u ) with T ( , . . . , tx ) li = 1 , \ T ( , . . . , txllo S
1 .
Nelson Dunford, Jacob T. Schwartz. 11 LEMMA . Let ( S , E , u ) be a positive
measure space and let { T ( 4 , . . . , tk ) , t , . . . , tx > 0 } be a strongly measurable
semi - group of operators in L ( S , E , u ) with T ( , . . . , tx ) li = 1 , \ T ( , . . . , txllo S
1 .
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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 spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect 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