## Linear Operators: General theory |

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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

u ( PE ) = II Ma ( EQ ) , del de for every elementary set Paed E , in S . Proof . We

will first show that u is

the stated value on elementary sets . For each a let Moe hy be set functions on

En ...

u ( PE ) = II Ma ( EQ ) , del de for every elementary set Paed E , in S . Proof . We

will first show that u is

**unique**. Let 2 be another additive set function on £ , withthe stated value on elementary sets . For each a let Moe hy be set functions on

En ...

Page 516

42 Show that in Exercise 38 the set function u is

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

each fe B ( S ) . 43 Show that in Exercise 39 the measure u is

positive ...

42 Show that in Exercise 38 the set function u is

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

each fe B ( S ) . 43 Show that in Exercise 39 the measure u is

**unique**up to apositive ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 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 Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential 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 mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero