## Linear Operators, Part 1 |

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Page 136

There is a

o - field containing a given family of sets . ... additive non - negative extension to

the o - field determined by E . If u is o - finite on Ethen this extension is

There is a

**uniquely**determined smallest field and a**uniquely**determined smallesto - field containing a given family of sets . ... additive non - negative extension to

the o - field determined by E . If u is o - finite on Ethen this extension is

**unique**.Page 202

We will first show that u is

with the stated value on elementary sets . For each a let uz , az be set functions

on & defined by the formulas My ( Ex ) = u ( E , XS . z . ) , 17 ( Ex ) = 2 ( EnXS . q ) ,

E ...

We will first show that u is

**unique**. Let 2 be another additive set function on £ ,with the stated value on elementary sets . For each a let uz , az be set functions

on & defined by the formulas My ( Ex ) = u ( E , XS . z . ) , 17 ( Ex ) = 2 ( EnXS . q ) ,

E ...

Page 516

43 Show that in Exercise 39 the measure u is

factor if and only if n - 1 X o / ( s ) ) converges uniformly to a constant for each f €

C ( S ) . 44 Let S be a compact metric space , and $ : S — S a mapping such that

e ...

43 Show that in Exercise 39 the measure u is

**unique**up to a positive constantfactor if and only if n - 1 X o / ( s ) ) converges uniformly to a constant for each f €

C ( S ) . 44 Let S be a compact metric space , and $ : S — S a mapping such that

e ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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

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