## Linear Operators, Part 1 |

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Page 136

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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 142

It follows from Theorem 14 that u has a regular countably additive

the o - field of all Borel sets in [ a , b ] . The restriction of this

field of Borel subsets of ( a , b ) is called the Radon or Borel - Stieltjes measure in

( a ...

It follows from Theorem 14 that u has a regular countably additive

**extension**tothe o - field of all Borel sets in [ a , b ] . The restriction of this

**extension**to the o -field of Borel subsets of ( a , b ) is called the Radon or Borel - Stieltjes measure in

( a ...

Page 143

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

o - field { * is known as the Lebesgue

and the measure space ( S , 2 * , u ) is the Lebesgue

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

**extension**of u . Theo - field { * is known as the Lebesgue

**extension**( relative to u ) of the o - field E ,and the measure space ( S , 2 * , 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 |

Algebraic Preliminaries | 34 |

Copyright | |

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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 obtained 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