Linear Operators: General theory |
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Page 100
... null set . This is done in the following definition . 11 DEFINITION . Let μ be an additive set function defined on a field of subsets of a set S. A subset N of S is said to be a μ - null set if v * ( μ , N ) 0 , where * is the extension ...
... null set . This is done in the following definition . 11 DEFINITION . Let μ be an additive set function defined on a field of subsets of a set S. A subset N of S is said to be a μ - null set if v * ( μ , N ) 0 , where * is the extension ...
Page 147
... set is a null set if and only if it is a subset of some measurable set F such that v ( μ , F ) = 0 . PROOF . If E is a null set , then v * ( μ , E ) = O and there are mea- surable sets En containing E with v ( u , En ) < 1 / n . Thus the ...
... set is a null set if and only if it is a subset of some measurable set F such that v ( μ , F ) = 0 . PROOF . If E is a null set , then v * ( μ , E ) = O and there are mea- surable sets En containing E with v ( u , En ) < 1 / n . Thus the ...
Page 213
... null set and 2 ( Q ) < rμ ( Q ) . ( b ) If for each p in a set ACG λ ( C ) > r , lim sup μ ( C ) → o μ ( C ) then each neighborhood of A contains a Borel set B such that A - B is a μ - null set and λ ( B ) > rμ ( B ) . PROOF . To prove ...
... null set and 2 ( Q ) < rμ ( Q ) . ( b ) If for each p in a set ACG λ ( C ) > r , lim sup μ ( C ) → o μ ( C ) then each neighborhood of A contains a Borel set B such that A - B is a μ - null set and λ ( B ) > rμ ( B ) . PROOF . To prove ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ