## Linear Operators: General theory |

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Page 132

Q.E.D. 5.

defined on a field ... This

theorems , of importance in later applications , will be discussed in this section .

Finally , it ...

Q.E.D. 5.

**Extensions**of Set Functions A given countably additive set functiondefined on a field ... This

**extension**theorem of Hahn and similar**extension**theorems , of importance in later applications , will be discussed in this section .

Finally , it ...

Page 136

( Hahn

set function u on a field E has a countably additive non - negative

the o - field determined by E. If u is o - finite on Ethen this

( Hahn

**extension**) Every countably additive non - negative extended real valuedset function u on a field E has a countably additive non - negative

**extension**tothe o - field determined by E. If u is o - finite on Ethen this

**extension**is unique .Page 143

Then the function u with domain { * is known as the Lebesgue

The o - field * is known as the Lebesgue

, and the measure space ( S , 3 * , u ) is the Lebesgue

Then the function u with domain { * is known as the Lebesgue

**extension**of u .The o - field * is known as the Lebesgue

**extension**( relative to u ) of the o - field E, and the measure space ( S , 3 * , u ) is the Lebesgue

**extension**of the measure ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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### Common terms and phrases

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