## Linear Operators, Part 1 |

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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 .

Page 96

A

additive if uld ) = 0 and u ( A , U AZ . . . U An ) = u ( A ) + u ( A2 ) + . . . tu ( An ) , for

every finite family { AL , . . . , An } of disjoint subsets of T whose union is in τ . For

an ...

A

**set function**u defined on a family r of sets is said to be additive or finitelyadditive if uld ) = 0 and u ( A , U AZ . . . U An ) = u ( A ) + u ( A2 ) + . . . tu ( An ) , for

every finite family { AL , . . . , An } of disjoint subsets of T whose union is in τ . For

an ...

Page 184

We shall denote by & the family of all sets in the Cartesian product S = Six . . .

XSn of the form E = EX . . . x En , where Eie & i . 1 LEMMA . For i = 1 , . . . , n , let

Mibe a finitely additive complex valued

subsets of ...

We shall denote by & the family of all sets in the Cartesian product S = Six . . .

XSn of the form E = EX . . . x En , where Eie & i . 1 LEMMA . For i = 1 , . . . , n , let

Mibe a finitely additive complex valued

**set function**defined on a field & ; ofsubsets of ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 other sections not shown

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint 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 obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero