## Linear Operators: General theory |

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Page 126

**Countably Additive**Set Functions The basis for the present section is a**countably additive**set function defined on a o - field of subsets of a set . In this case the results of the preceding sections can be considerably extended .Page 132

This lemma shows that if each member of a generalized sequence { 2 , } of finite ,

This lemma shows that if each member of a generalized sequence { 2 , } of finite ,

**countably additive**measures is u - continuous and if lim , 2 ( E ) = 2 ( E ) , E £ E , where 2 is also a finite ,**countably additive**measure , then 2 is ...Page 136

( Hahn extension ) Every

( Hahn extension ) Every

**countably additive**non - negative extended real valued set function u on a field E has a**countably additive**non - negative extension to the o - field determined by £ . If u is o - finite on then this extension ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero