## Linear Operators: General theory |

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Page 143

The o - field * is known as the Lebesgue extension ( relative to u ) of the o - field E , and the

The o - field * 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 the**measure**space ( S , E , u ) . Expressions such as u - simple function ...Page 186

Thus , the field Ø of Lemma 1 is a o - field and the

Thus , the field Ø of Lemma 1 is a o - field and the

**measure**и M of Lemma 1 is countably additive on Ø . Consequently , the restriction of u to the o - field generated by E is a**measure**with the desired properties .Page 188

It is clear that the

It is clear that the

**measure**has the property n n MC PE ; ) = I1 M : ( E . ) , Ε , ε Σ . , and it follows from the uniqueness part of Theorem 2 for each of the spaces ( EG ) , ) , ( ) ) , that u is the unique countably additive**measure**...### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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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