Linear Operators, Part 1 |
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Page 95
Finitely Additive Set Functions In contrast to the terms real function , complex
function , etc. , where the adjectives real and ... of the function , the term set
function is commonly used in mathematics for a function whose domain is a
family of sets .
Finitely Additive Set Functions In contrast to the terms real function , complex
function , etc. , where the adjectives real and ... of the function , the term set
function is commonly used in mathematics for a function whose domain is a
family of sets .
Page 96
2 DEFINITION . A set function u defined on a family t of sets is said to be additive
or finitely additive if uld ) = 0 and M ( A , U AZ ... U An ) = u ( 41 ) + u ( A2 ) + ... + u
( An ) , for every finite family { Aj , ... An } of disjoint subsets of t whose union is in ...
2 DEFINITION . A set function u defined on a family t of sets is said to be additive
or finitely additive if uld ) = 0 and M ( A , U AZ ... U An ) = u ( 41 ) + u ( A2 ) + ... + u
( An ) , for every finite family { Aj , ... An } of disjoint subsets of t whose union is in ...
Page 837
A on Abel summability , of series , II.4.42 ( 76 ) Abelian group , ( 34 ) Absolutely
continuous functions , definition , IV.2.22 ( 242 ) set function . ( See Continuous
set function and Set function ) space of , additional properties , IV.15 ( 378 ) ...
A on Abel summability , of series , II.4.42 ( 76 ) Abelian group , ( 34 ) Absolutely
continuous functions , definition , IV.2.22 ( 242 ) set function . ( See Continuous
set function and Set function ) space of , additional properties , IV.15 ( 378 ) ...
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Contents
Preliminary Concepts | 1 |
The VitaliHahnSaks Theorem and Spaces of Measures | 7 |
B Topological Preliminaries | 10 |
Copyright | |
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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