Linear Operators: General theory |
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12 LEMMA . Let ( R , ER , 0 ) be the product of finite measure spaces ( S , L , \ ) and ( T , E7 , 2 ) . Let E be a g - null set in R. Then for 2 - almost all t , the set E ( t ) = { $ [ $ , t ] € E } is a u - null set . PROOF .
12 LEMMA . Let ( R , ER , 0 ) be the product of finite measure spaces ( S , L , \ ) and ( T , E7 , 2 ) . Let E be a g - null set in R. Then for 2 - almost all t , the set E ( t ) = { $ [ $ , t ] € E } is a u - null set . PROOF .
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11 LEMMA . Let ( S , E , u ) be a positive measure space and let { T ( 4 , ... , tk ) , ty , ... , tx > 0 } be a strongly measurable semi - group of operators in Lj ( S , E , u ) with \ T ( tj , ... , tx ) 1 = 1 , T ( 41 , ... , tko s 1 ...
11 LEMMA . Let ( S , E , u ) be a positive measure space and let { T ( 4 , ... , tk ) , ty , ... , tx > 0 } be a strongly measurable semi - group of operators in Lj ( S , E , u ) with \ T ( tj , ... , tx ) 1 = 1 , T ( 41 , ... , tko s 1 ...
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2 v 1 Jo 26 - V + p V t - * - » ) So Va 2y2 e - Vt ammad dx 21 Jo X - 3 / 2 2 e dy 2e - v't --- A - évt , V1 which proves ( * ) and completes the proof of the lemma . Q.E.D. We shall now state and prove the lemma referred to as CPx .
2 v 1 Jo 26 - V + p V t - * - » ) So Va 2y2 e - Vt ammad dx 21 Jo X - 3 / 2 2 e dy 2e - v't --- A - évt , V1 which proves ( * ) and completes the proof of the lemma . Q.E.D. We shall now state and prove the lemma referred to as CPx .
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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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