## Linear Operators: General theory |

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Page 106

The functions totally u - measurable on S , or , if u is understood , totally

measurable on S are the functions in the closure

simple functions . If for every E in { with v ( u , E ) < , the product met of s with the

characteristic ...

The functions totally u - measurable on S , or , if u is understood , totally

measurable on S are the functions in the closure

**TM**(**S**) in F ( S ) of the u -simple functions . If for every E in { with v ( u , E ) < , the product met of s with the

characteristic ...

Page 329

The Space

its subsets , and a scalar valued countably additive set function u on £ . The

symbol

which ...

The Space

**TM**(**S**, E , u ) We shall be concerned here with a set S , a o - field ofits subsets , and a scalar valued countably additive set function u on £ . The

symbol

**TM**(**S**, E , j ) will be used for the set of all scalar valued functions on Swhich ...

Page 333

Suppose that ( i ) for each x in X sup T2 ( x , s ) ] < 0 , αελ , almost everywhere on

S ; and ( ii ) for each x in a set dense in X lim sup \ T2 ( x , s ) — T ( x ...

Consequently we suppose that u is a finite measure , and so

S , E , u ) .

Suppose that ( i ) for each x in X sup T2 ( x , s ) ] < 0 , αελ , almost everywhere on

S ; and ( ii ) for each x in a set dense in X lim sup \ T2 ( x , s ) — T ( x ...

Consequently we suppose that u is a finite measure , and so

**TM**(**S**, E , u ) = M (S , E , u ) .

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

algebra analytic applied arbitrary assumed B-space ba(S Borel bounded called Chapter clear closed compact complex condition Consequently constant contains continuous functions converges Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hausdorff Hence Hilbert space identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear space mapping Math means measure space neighborhood norm obtained operator positive measure preceding projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero