## Linear Operators: General theory |

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Page 96

2 DEFINITION . A

or finitely

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

2 DEFINITION . A

**set function**u defined on a family t of**sets**is said to be**additive**or finitely

**additive**if uld ) = 0 and u ( A4U Ag . . . U An ) = M ( A2 ) + u ( A2 ) + . . . tu( An ) , for every finite family { A , , . . . , An } of disjoint subsets of t whose union ...

Page 97

The total variation v ( u ) of an

dominates u in the sense that vlu , E ) > lu ( E ) ... comprehension of Definition 4

below by proving that v ( u ) is the smallest of the non - negative

The total variation v ( u ) of an

**additive set function**u is important because itdominates u in the sense that vlu , E ) > lu ( E ) ... comprehension of Definition 4

below by proving that v ( u ) is the smallest of the non - negative

**additive set****functions**...Page 126

Countably

results of the preceding sections can be considerably extended . 1 DEFINITION .

Countably

**Additive Set Functions**The basis for the present section is a countably**additive set function**defined on a o - field of subsets of a set . In this case theresults of the preceding sections can be considerably extended . 1 DEFINITION .

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

algebra analytic applied arbitrary assumed B-space ba(S Borel bounded called Chapter clear closed compact complex condition Consequently constant contains continuous functions converges Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hausdorff Hence Hilbert space identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear space mapping Math means measure space neighborhood norm obtained operator positive measure preceding projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero