## Linear Operators, Part 1 |

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Page 95

Finitely Additive

function , etc. , where the adjectives real and ... of the function , the term

family of sets .

Finitely Additive

**Set Functions**In contrast to the terms real function , complexfunction , etc. , where the adjectives real and ... of the function , the term

**set****function**is commonly used in mathematics for a function whose domain is afamily of sets .

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2 DEFINITION . A

or finitely additive if uld ) = 0 and M ( A , U AZ ... U An ) = u ( 41 ) + u ( A2 ) + ... + u

( An ) , for every finite family { Aj , ... An } of disjoint subsets of t whose union is in ...

2 DEFINITION . A

**set function**u defined on a family t of sets is said to be additiveor finitely additive if uld ) = 0 and M ( A , U AZ ... U An ) = u ( 41 ) + u ( A2 ) + ... + u

( An ) , for every finite family { Aj , ... An } of disjoint subsets of t whose union is in ...

Page 837

A on Abel summability , of series , II.4.42 ( 76 ) Abelian group , ( 34 ) Absolutely

continuous functions , definition , IV.2.22 ( 242 )

A on Abel summability , of series , II.4.42 ( 76 ) Abelian group , ( 34 ) Absolutely

continuous functions , definition , IV.2.22 ( 242 )

**set function**. ( See Continuous**set function**and**Set function**) space of , additional properties , IV.15 ( 378 ) ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero