Linear Operators: General theory |
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Page 136
... unique . μ 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 Zo determined by 2. Suppose that is another such extension . If μ is o - finite on to prove ...
... unique . μ 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 Zo determined by 2. Suppose that is another such extension . If μ is o - finite on to prove ...
Page 202
... unique . This uniqueness argument suggests how we may prove the existence of μ . For each finite set л in A there is , by Lemma 1 , a unique additive set function μ , defined on Σ , which has the property - ( PE ) II Ma ( Ea ) , аел αεπ ...
... unique . This uniqueness argument suggests how we may prove the existence of μ . For each finite set л in A there is , by Lemma 1 , a unique additive set function μ , defined on Σ , which has the property - ( PE ) II Ma ( Ea ) , аел 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 u defined for the Borel subsets of S such that μ ( p - 1 ( E ) ) = μ ( 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 u defined for the Borel subsets of S such that μ ( p - 1 ( E ) ) = μ ( E ) for each Borel ...
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