Linear Operators: General theory |
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Page 126
Nelson Dunford, Jacob T. Schwartz. 4. Countably Additive Set Functions The basis for the present section is a countably additive set func- tion defined on a σ - field of subsets of a set . In this case the results of the preceding ...
Nelson Dunford, Jacob T. Schwartz. 4. Countably Additive Set Functions The basis for the present section is a countably additive set func- tion defined on a σ - field of subsets of a set . In this case the results of the preceding ...
Page 132
... countably additive measures is μ - continuous and if lim 2 ( E ) = 2 ( E ) , Ee , where 2 is also a finite , countably additive measure , then λ is also μ - continuous . 1 , 2 , .... are It follows easily from Definition 12 that if 2 ...
... countably additive measures is μ - continuous and if lim 2 ( E ) = 2 ( E ) , Ee , where 2 is also a finite , countably additive measure , then λ is also μ - continuous . 1 , 2 , .... are It follows easily from Definition 12 that if 2 ...
Page 136
... countably additive non - neg- ative extended real valued set function μ on a field Σ has a countably additive non - negative extension to the o - field determined by Σ . If μ is o - finite on then this extension is unique . PROOF ...
... countably additive non - neg- ative extended real valued set function μ on a field Σ has a countably additive non - negative extension to the o - field determined by Σ . If μ is o - finite on then this extension is unique . PROOF ...
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 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ