## 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

u ( P Eq ) = II ua ( EQ ) , αεΑ αεΑ for every elementary set Paca Eq in S . Proof .

We will first show that u is

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

on ...

u ( P Eq ) = II ua ( EQ ) , αεΑ αεΑ for every elementary set Paca Eq in S . Proof .

We will first show that u is

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

on ...

Page 516

42 Show that in Exercise 38 the set function u is

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

each fe 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 & n = 0 / ( $ { ( s ) ) converges uniformly to a constant for

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

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

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero