## Linear Operators: General theory |

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Page 136

There is a

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

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 Mz hq be set functions on En defined by the formulas My ( Ex ) = u ( EnXS ) , 27 ( Ex ) ...Page 516

42 Show that in Exercise 38 the set function u is

42 Show that in Exercise 38 the set function u is

**unique**up to a positive constant factor if and only if n - 1 X = 0 / ( $ ' ( s ) ) converges uniformly to a constant for each f € B ( S ) . 43 Show that in Exercise 39 the measure u is ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### 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 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 u-measurable uniform uniformly unique unit valued vector weak weakly compact zero