## Linear Operators: General theory |

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Page 106

The

characteristic

The

**functions**totally u -**measurable**on S , or , if u is understood , totally**measurable**on S are the**functions**in the ... the product met of s with thecharacteristic

**function**xe of E is totally**measurable**, the**function**f is said to be u -**measurable**or , if ...Page 179

Let ( S , E , u ) be a positive measure space , f a nonnegative u -

negative a -

) h ( s ) ...

Let ( S , E , u ) be a positive measure space , f a nonnegative u -

**measurable****function**defined on S and 2 ( E ) = f ( s ) u ... If we let H be the class of all non -negative a -

**measurable functions**h for which the equation sh ( s ) a ( ds ) = ( f ( s) h ( s ) ...

Page 180

Theorem 4 applies in the same manner to prove the 2 - integrability of g when it is

assumed that tg is u - integrable . Since every real

difference of two nonnegative

Theorem 4 applies in the same manner to prove the 2 - integrability of g when it is

assumed that tg is u - integrable . Since every real

**measurable function**is thedifference of two nonnegative

**measurable functions**, it follows from the theorem ...### What people are saying - Write a review

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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