Linear Operators: General theory |
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Page 136
... unique . PROOF . Theorem 4 and Lemma 5 show that the outer measure μ is one non - negative countably additive extension of μ to the o - field E determined by E. Suppose that μ1 is another such extension . If μ is o - finite on Σ to ...
... unique . PROOF . Theorem 4 and Lemma 5 show that the outer measure μ is one non - negative countably additive extension of μ to the o - field E determined by E. Suppose that μ1 is another such extension . If μ is o - finite on Σ to ...
Page 202
... unique . This uniqueness argument suggests how we may prove the existence of u . For each finite set л in A there is , by Lemma 1 , a unique additive set function u , defined on 2 which has the property ( PE ) = II μa ( Ex ) , αεπ αεπ ...
... unique . This uniqueness argument suggests how we may prove the existence of u . For each finite set л in A there is , by Lemma 1 , a unique additive set function u , defined on 2 which has the property ( PE ) = II μa ( Ex ) , αεπ αεπ ...
Page 516
... unique up to a positive constant factor if and only if n - 1 ( p ( s ) ) converges uniformly to a constant for each ... unique regular measure μ defined for the Borel subsets of S such that μ ( 4 - 1 ( E ) ) = u ( E ) for each Borel ...
... unique up to a positive constant factor if and only if n - 1 ( p ( s ) ) converges uniformly to a constant for each ... unique regular measure μ defined for the Borel subsets of S such that μ ( 4 - 1 ( E ) ) = u ( E ) for each Borel ...
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