## Linear Operators, Part 1 |

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Page 136

6 LEMMA . There is a

determined smallest o - field containing a given family of sets . PROOF . There is

at least one field , namely the field of all subsets of S , which contains a given

family t .

6 LEMMA . There is a

**uniquely**determined smallest field and a**uniquely**determined smallest o - field containing a given family of sets . PROOF . There is

at least one field , namely the field of all subsets of S , which contains a given

family t .

Page 202

We will first show that u is

with the stated value on elementary sets . For each a let Miq , 2 , be set functions

on En defined by the formulas M7 ( EZ ) = u ( E X Szi ) , 2 , ( ER ) = 2 ( E , XS .

We will first show that u is

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

on En defined by the formulas M7 ( EZ ) = u ( E X Szi ) , 2 , ( ER ) = 2 ( E , XS .

Page 516

43 Show that in Exercise 39 the measure u is

factor if and only if n - 1 = 0 / ( $ ' ( s ) ) converges uniformly to a constant for each

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

that ...

43 Show that in Exercise 39 the measure u is

**unique**up to a positive constantfactor if and only if n - 1 = 0 / ( $ ' ( s ) ) converges uniformly to a constant for each

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

that ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### 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 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 isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space 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