Linear Operators: General theory |
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Page 100
... null set . This is done in the following definition . μ 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 u - null set if v * ( u , N ) = 0 , where v * is the ...
... null set . This is done in the following definition . μ 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 u - null set if v * ( u , N ) = 0 , where v * is the ...
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 * ( u , E ) - O and there are mea- surable sets E , containing E with v ( u , E ) < 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 * ( u , E ) - O and there are mea- surable sets E , containing E with v ( u , E ) < 1 / n . Thus the ...
Page 213
... null set and 2 ( Q ) < rμ ( Q ) . ( b ) If for each p in a set ACG 2 ( C ) r , lim sup μ ( C ) → 0 μ ( C ) then each neighborhood of A contains a Borel set B such that A - B is a u - 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 2 ( C ) r , lim sup μ ( C ) → 0 μ ( C ) then each neighborhood of A contains a Borel set B such that A - B is a u - null set and λ ( B ) > rμ ( B ) . PROOF . To prove ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ