## Linear Operators, Part 1 |

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Page 131

Let a , u be finitely additive set functions defined on a field E. Then 2 is said to be

continuous with respect to u or simply l - continuous , if lim 2 ( E ) = 0 . v ( u , E ) -

> 0 The function 2 is said to be u -

Let a , u be finitely additive set functions defined on a field E. Then 2 is said to be

continuous with respect to u or simply l - continuous , if lim 2 ( E ) = 0 . v ( u , E ) -

> 0 The function 2 is said to be u -

**singular**if there is a set E , e { such that v ( u ...Page 132

It follows easily from Definition 12 that if 2. , ano n = 1 , 2 , ... , are finite countably

additive measures with an ( E ) → ( E ) , E € E , and if in , n = 1 , 2 , ... are u -

( S ...

It follows easily from Definition 12 that if 2. , ano n = 1 , 2 , ... , are finite countably

additive measures with an ( E ) → ( E ) , E € E , and if in , n = 1 , 2 , ... are u -

**singular**, then 2 is also u -**singular**. THEOREM . ( Lebesgue decomposition ) Let( S ...

Page 855

Simple function ( s ) , definition , III.2.9 ( 105 ) density in Lp , isp < 0 of , III.3.8 (

125 ) , I11.8.3 ( 167 ) , III.9.46 ( 174 ) Simple Jordan curve , ( 225 )

element in a ring , ( 40 ) non -

definition ...

Simple function ( s ) , definition , III.2.9 ( 105 ) density in Lp , isp < 0 of , III.3.8 (

125 ) , I11.8.3 ( 167 ) , III.9.46 ( 174 ) Simple Jordan curve , ( 225 )

**Singular**element in a ring , ( 40 ) non -

**singular**operator , ( 45 )**Singular**set function ,definition ...

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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

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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