## Linear Operators: General theory |

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Page 95

Finitely Additive

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

commonly used in mathematics for a function whose domain is a family of sets .

Finitely Additive

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

**set function**iscommonly used in mathematics for a function whose domain is a family of sets .

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INTEGRATION AND

defined on a family of sets is said to be additive or finitely additive it u = ) and u (

A4U Ag . . . U An ) = u ( A1 ) + u ( A2 ) + . . . + ( An ) , for every finite family { A , , . . .

INTEGRATION AND

**SET FUNCTIONS**III . 1 . 2 2 DEFINITION . A**set function**udefined on a family of sets is said to be additive or finitely additive it u = ) and u (

A4U Ag . . . U An ) = u ( A1 ) + u ( A2 ) + . . . + ( An ) , for every finite family { A , , . . .

Page 131

Let 2 , u be finitely additive

continuous with respect to u or simply M - continuous , if lim 2 ( E ) = 0 . olu , E ) →

0 . The function 2 is said to be u - singular if there is a set E . c & such that v ( u ...

Let 2 , u be finitely additive

**set functions**defined on a field E . Then 2 is said to becontinuous with respect to u or simply M - continuous , if lim 2 ( E ) = 0 . olu , E ) →

0 . The function 2 is said to be u - singular if there is a set E . c & such that v ( u ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

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