## Linear Operators, Part 1 |

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Page 136

There is a uniquely determined smallest

o -

namely the

intersection ...

There is a uniquely determined smallest

**field**and a uniquely determined smallesto -

**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 . Theintersection ...

Page 166

If we put E ( E ) = { F € E F C E } it is clear that E ( E ) is a

that E ( E ) is the family of all sets AE , A € X , and that if Eis a o -

is a o -

If we put E ( E ) = { F € E F C E } it is clear that E ( E ) is a

**field**of subsets of E , andthat E ( E ) is the family of all sets AE , A € X , and that if Eis a o -

**field**, then E ( E )is a o -

**field**. Σ ( Ε ) is called the restriction of Σ to E. If Σ , is a**field**, Εε Σ . , and ...Page 201

The symbol will be used for the o -

of all sets in S of the form S , XER with Ε ε Σ will be denoted by Σπ . 18 LEMMA .

For each a the family In is a

The symbol will be used for the o -

**field**of sets in S determined by Ey . The familyof all sets in S of the form S , XER with Ε ε Σ will be denoted by Σπ . 18 LEMMA .

For each a the family In is a

**field**of sets in S and Ei = U241 . PROOF .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero