## Linear Operators: General theory |

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Page 95

Finitely Additive Set

Finitely Additive Set

**Functions**In contrast to the terms real**function**, complex**function**, etc. , where the adjectives ... Thus , even if the integration process is**defined**with respect to a scalar valued set**function**u , the resulting ...Page 98

The total variation of an additive set

The total variation of an additive set

**function defined**on a field of subsets of a set S is also additive on E. PROOF . Let { A ; } be a finite sequence of disjoint sets in with ACEUF where E , F € £ and EF = $ .Page 101

In these sections † will be a vector valued

In these sections † will be a vector valued

**function defined**on a set S and u a finitely additive set**function defined**on a field of subsets of S. It will not be assumed that u is bounded . Thus the basis for this material is a fixed ...### What people are saying - Write a review

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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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### 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