## Linear Operators: General theory |

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Page 132

This

This

**extension**theorem of Hahn and similar**extension**theorems , of importance in later applications , will be discussed in this section . Finally , it is shown how the**extension**theorems may be used to construct the classical measures ...Page 136

( Hahn

( Hahn

**extension**) Every countably additive non - negative extended real valued set function u on a field & has a countably additive non - negative**extension**to the o - field determined by E. If u is o - finite on then this**extension**is ...Page 143

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

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

**extension**of u . The o - field 2 * is known as the Lebesgue**extension**( relative to u ) of the o - field E , and the measure space ( S , * , u ) is the Lebesgue**extension**of ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero