## 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 E of subsets of a set S is also additive on E. Proof . Let { A ; } be a finite sequence of disjoint sets in E with A ; CEUF where E , F £ and EF = $ .Page 182

Q.E.D. Note : The unique u - integrable

Q.E.D. Note : The unique u - integrable

**function**of Theorems 2 and 7 is called the Radon - Nikodım derivative of 2 with respect to u and is often denoted by da / du . Thus da du is**defined**u - almost everywhere by the formula lai 2 ( E ) ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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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