## 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 | |

80 other sections not shown

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