## 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 { 4 ; } be a finite sequence of disjoint sets in with ACEUF where E , FE and EF = $ .Page 101

SES SES If for some null set N the restriction of f to S - N has its values in a separable set , the

SES SES If for some null set N the restriction of f to S - N has its values in a separable set , the

**function**f is said to be i - essentially separably valued . Expressions such as essentially compact valued are**defined**similarly .### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

Acad 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 semi-group 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