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 ) , EeΣ , where λ 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 ) , EeΣ , where λ 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 . μ 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 . μ 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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ