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Page 143
Then the function u with domain * is known as the Lebesgue extension of u . The
o - field { * is known as the Lebesgue extension ( relative to u ) of the o - field E ,
and the measure space ( S , 2 * , u ) is the Lebesgue extension of the measure ...
Then the function u with domain * is known as the Lebesgue extension of u . The
o - field { * is known as the Lebesgue extension ( relative to u ) of the o - field E ,
and the measure space ( S , 2 * , u ) is the Lebesgue extension of the measure ...
Page 218
9 DEFINITION . Let f be a vector valued Lebesgue integrable function defined on
an open set in Euclidean n - space . The set of all points p at which lim - H ( q ) - 1
( Plu ( dq ) = 0 M ( C ) 0 ( C ) JC is called the Lebesgue set of the function f .
9 DEFINITION . Let f be a vector valued Lebesgue integrable function defined on
an open set in Euclidean n - space . The set of all points p at which lim - H ( q ) - 1
( Plu ( dq ) = 0 M ( C ) 0 ( C ) JC is called the Lebesgue set of the function f .
Page 223
5 Let h be a function of bounded variation on the interval ( a , b ) and continuous
on the right . Let g be a function defined on ( a , b ) such that the Lebesgue -
Stieltjes integral I = Sag ( s ) dh ( s ) exists . Let f be a continuous increasing
function ...
5 Let h be a function of bounded variation on the interval ( a , b ) and continuous
on the right . Let g be a function defined on ( a , b ) such that the Lebesgue -
Stieltjes integral I = Sag ( s ) dh ( s ) exists . Let f be a continuous increasing
function ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero