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 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