## Linear Operators: General theory |

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Page 106

The functions totally

The functions totally

**u**-**measurable**on S , or , if**u**is understood , 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 ) < oo , the product Xet ...Page 149

If ( S , E ,

If ( S , E ,

**u**) is a measure space , then a function | on S to a B - space X is**u**-**measurable**if and only if ( i ) for every**measurable**set F with v (**u**, F ) < oo , the function | is j - essentially separably valued on F , and ( ii ) ...Page 178

Thus we may and shall assume that v (

Thus we may and shall assume that v (

**u**, F ) < oo and v ( 2 , F ) < oo . Since g is 2 -**measurable**there is a sequence { n } of simple functions converging to g ( s ) for every s in F except on a set ECF with v ( 1 , E ) = 0 ( by ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain 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 mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero