## 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 .

Page 96

Nelson Dunford, Jacob T. Schwartz. the med 96 III . INTEGRATION AND

is said to be additive or finitely additive it u = ) and u ( A4U Ag . . . U An ) = u ( A1 )

+ ...

Nelson Dunford, Jacob T. Schwartz. the med 96 III . INTEGRATION AND

**SET****FUNCTIONS**III . 1 . 2 2 DEFINITION . A**set function**u defined on a family of setsis said to be additive or finitely additive it u = ) and u ( A4U Ag . . . U An ) = u ( A1 )

+ ...

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 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

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