Linear Operators: General theory |
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Page 132
... countably additive measures is μ - continuous and if lim λ ( E ) = 2 ( E ) , Ee , where λ is also a finite , countably additive measure , then λ is also u - continuous . It follows easily from Definition 12 that if 2 , λn , n = 1 , 2 ...
... countably additive measures is μ - continuous and if lim λ ( E ) = 2 ( E ) , Ee , where λ is also a finite , countably additive measure , then λ is also u - continuous . It follows easily from Definition 12 that if 2 , λn , n = 1 , 2 ...
Page 136
... countably additive non - neg- ative extended real valued set function u on a field E 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 u on a field E 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 ...
Page 405
... countably additive ; the subset ↳1⁄2 of s is of μ - measure zero ; the measure is not countably additive . In fact , using the relation between μ∞ and μ , it is easy to see that l is the union of a countable family of u - null sets ...
... countably additive ; the subset ↳1⁄2 of s is of μ - measure zero ; the measure is not countably additive . In fact , using the relation between μ∞ and μ , it is easy to see that l is the union of a countable family of u - null sets ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz 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