## Linear Operators, Part 1 |

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12

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 . By

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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 . By

**Lemma**11 it may be assumed that ...Page 697

Nelson Dunford, Jacob T. Schwartz. 11

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 .

Nelson Dunford, Jacob T. Schwartz. 11

**LEMMA**. Let ( S , E , u ) be a positivemeasure 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**lemma**is ...### 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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