## Linear Operators: General theory |

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Page 106

The functions 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 let off with the

characteristic ...

The functions totally

**u**-**measurable**on S , or , if u is understood , totallymeasurable 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 let off with the

characteristic ...

Page 119

( b ) the function g defined by g ( s ) = f ( s ) if s & S + US- , g ( s ) = 0 if s eS + US- ,

is

extended real - valued ) which is defined only on the complement of a u - null set

NĒS .

( b ) the function g defined by g ( s ) = f ( s ) if s & S + US- , g ( s ) = 0 if s eS + US- ,

is

**u**-**measurable**. Next suppose that we consider a function | ( vector orextended real - valued ) which is defined only on the complement of a u - null set

NĒS .

Page 178

But old , E ) = 1 , \ | ( s ) [ 0 ( u , ds ) so that f ( s ) = 0 for s in E except in a set A with

v ( u , A ) = 0 . Thus 8n ( $ ) { ( 8 ) ▻g ( s ) [ ( 8 ) , 8€ F - A , and Corollary 6 . 14

shows that Xefg is

...

But old , E ) = 1 , \ | ( s ) [ 0 ( u , ds ) so that f ( s ) = 0 for s in E except in a set A with

v ( u , A ) = 0 . Thus 8n ( $ ) { ( 8 ) ▻g ( s ) [ ( 8 ) , 8€ F - A , and Corollary 6 . 14

shows that Xefg is

**u**-**measurable**. Conversely , let fg be**u**-**measurable**, and let...

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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 Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero