## Linear Operators: General theory |

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Page 106

If for every E in { with v ( u , E ) < oo , the product met of | with the characteristic

function Xe of E is totally measurable , the function f is said to be

or , if u is understood , simply measurable . A set A is measurable if Xa is ...

If for every E in { with v ( u , E ) < oo , the product met of | with the characteristic

function Xe of E is totally measurable , the function f is said to be

**u**-**measurable**or , if u is understood , simply measurable . A set A is measurable if Xa is ...

Page 119

( b ) the function g defined by g ( s ) = f ( s ) if s & S + US- , g ( s ) = 0 if s eS + US- ,

is

extended real - valued ) which is defined only on the complement of a u - null set

NĒS .

( b ) the function g defined by g ( s ) = f ( s ) if s & S + US- , g ( s ) = 0 if s eS + US- ,

is

**u**-**measurable**. Next suppose that we consider a function | ( vector orextended real - valued ) which is defined only on the complement of a u - null set

NĒS .

Page 178

XF ( s ) / ( 8 ) g ( 8 ) = lim XF , ( s ) / ( 8 ) g ( 8 ) , SES , and the pointwise limit of a

sequence of

Xe.tg is

...

XF ( s ) / ( 8 ) g ( 8 ) = lim XF , ( s ) / ( 8 ) g ( 8 ) , SES , and the pointwise limit of a

sequence of

**measurable**functions is**measurable**, it will suffice then to show thatXe.tg is

**measurable**. Thus we may and shall assume that v (**u**, F ) < oo and v ( a...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 meaning measure space metric neighborhood norm operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero