## Linear Operators: General theory |

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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 then 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 then this extension is

**unique**.Page 202

We will first show that u is

the stated value on elementary sets . For each r let Mine debe set functions on En

defined by the formulas M ( EZ ) = u ( E , XSX ) , 2 , ( EZ ) = 2 ( E , XS , ) , Εε Σ ..

We will first show that u is

**unique**. Let 2 be another additive set function on withthe stated value on elementary sets . For each r let Mine debe set functions on En

defined by the formulas M ( EZ ) = u ( E , XSX ) , 2 , ( EZ ) = 2 ( E , XS , ) , Εε Σ ..

Page 516

42 Show that in Exercise 38 the set function u is

factor if and only if n - 1 2-1 ( * ' ( s ) ) converges uniformly to a constant for each

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

42 Show that in Exercise 38 the set function u is

**unique**up to a positive constantfactor if and only if n - 1 2-1 ( * ' ( s ) ) converges uniformly to a constant for each

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

**unique**up to a positive ...### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 operator positive measure problem Proc PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero