## 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 then 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 then this

**extension**is unique .Page 143

Then the function u with domain 2 * 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 2 * 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

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 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

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero