## Linear Operators: General theory |

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Finitely Additive

Finitely Additive

**Set Functions**In contrast to the terms real function , complex function , etc. , where the adjectives real and complex refer to the range of the function , the term**set function**is commonly used in mathematics for a ...Page 96

INTEGRATION AND

INTEGRATION AND

**SET FUNCTIONS**III.1.2 2 DEFINITION . A**set function**u defined on a family t of sets is said to be additive or finitely additive it ud O and u ( ALU AZ ... U An ) = u ( A1 ) + u ( A2 ) + .. tu ( An ) , for every finite ...Page 184

We shall denote by & the family of all sets in the Cartesian product S = S , X ... XSn of the form E = EX ... % . En , where Eie & i . 1 LEMMA . For i = 1 , ... , n , let u ; be a finitely additive complex valued

We shall denote by & the family of all sets in the Cartesian product S = S , X ... XSn of the form E = EX ... % . En , where Eie & i . 1 LEMMA . For i = 1 , ... , n , let u ; be a finitely additive complex valued

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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 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 Hilbert space 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