## Linear Operators: General theory |

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Page 136

There is a uniquely determined smallest field and a uniquely determined smallest

namely the field of all subsets of S , which contains a given family t .

There is a uniquely determined smallest field and a uniquely determined smallest

**o**-**field**containing a given family of sets . PROOF . There is at least one field ,namely the field of all subsets of S , which contains a given family t .

Page 143

Let u be a vector or extended real valued function on the

defined as in the preceding theorem . Then the function u with domain & * is

known as the Lebesgue extension of u . The

Lebesgue ...

Let u be a vector or extended real valued function on the

**o**-**field**and let 2 * bedefined as in the preceding theorem . Then the function u with domain & * is

known as the Lebesgue extension of u . The

**o**-**field**2 * is known as theLebesgue ...

Page 166

By the argument presented in the proof of Theorem 2.22 we may and shall

assume that In ( s ) 5 2 \ | ( 8 ) for all s in S. By the ... Σ ( Ε ) is called the restriction

of Σ to E. If Σ , is a field , Εε Σ . , and Σ is the

easily ...

By the argument presented in the proof of Theorem 2.22 we may and shall

assume that In ( s ) 5 2 \ | ( 8 ) for all s in S. By the ... Σ ( Ε ) is called the restriction

of Σ to E. If Σ , is a field , Εε Σ . , and Σ is the

**o**-**field**generated by { , then we caneasily ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains 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 Hilbert space 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