Linear Operators: General theory |
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Page 132
... countably additive measures is μ - continuous and if lim ̧ λ 。( E ) = 2 ( E ) , EeΣ , where λ is also a finite , countably additive measure , then λ is also μ - continuous . It follows easily from Definition 12 that if 2 , λn , n = 1 ...
... countably additive measures is μ - continuous and if lim ̧ λ 。( E ) = 2 ( E ) , EeΣ , where λ is also a finite , countably additive measure , then λ is also μ - continuous . It follows easily from Definition 12 that if 2 , λn , n = 1 ...
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 2. If u 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 2. If u is o - finite on then this extension is unique . PROOF ...
Page 405
... countably additive ; the subset 2 of 8 is of μ - measure zero ; the measure μ is not countably additive . In fact , using the relation between μ∞ and Ма it is easy to see that l is the union of a countable family of u - null sets ...
... countably additive ; the subset 2 of 8 is of μ - measure zero ; the measure μ is not countably additive . In fact , using the relation between μ∞ and Ма it is easy to see that l is the union of a countable family of u - null sets ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ