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 Zo determined by E. Suppose that is another such extension . If u is σ - finite on Σ to prove ...
... 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 Zo determined by E. Suppose that is another such extension . If u is σ - finite on Σ to prove ...
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 μ , defined on 2 , which has the property 九( PE ) = II Ma ( Ex ) , 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 μ , defined on 2 , which has the property 九( PE ) = II Ma ( Ex ) , αεπ ...
Page 516
... unique up to a positive constant factor if and only if n - 1 Σ = 1f ( p ( s ) ) converges uniformly to a constant ... unique regular measure u defined for the Borel subsets of S such that μ ( þ − 1 ( E ) ) = μ ( E ) for each Borel ...
... unique up to a positive constant factor if and only if n - 1 Σ = 1f ( p ( s ) ) converges uniformly to a constant ... unique regular measure u defined for the Borel subsets of S such that μ ( þ − 1 ( E ) ) = μ ( E ) for each Borel ...
Contents
Preliminary Concepts | 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 ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ