## Linear Operators: General theory |

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**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 .Page 697

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**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 ...Page 699

2 v 1 Jo 26 - V + p V t - * - » ) So Va 2y2 2 e dy 2e - v't which proves ( * ) and completes the proof of the

2 v 1 Jo 26 - V + p V t - * - » ) So Va 2y2 2 e dy 2e - v't which proves ( * ) and completes the proof of the

**lemma**. Q.E.D. We shall now state and prove the**lemma**referred to as CPx . For technical reasons occurring later the following ...### What people are saying - Write a review

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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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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero