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Page 145
... uniformly on S - E . The sequence { f } converges u - uniformly to the function f if for each & > 0 there is a set E e Σ such that v ( μ , E ) < ɛ and such that { f } converges uniformly to f on S - E . n It is clear that u - uniform ...
... uniformly on S - E . The sequence { f } converges u - uniformly to the function f if for each & > 0 there is a set E e Σ such that v ( μ , E ) < ɛ and such that { f } converges uniformly to f on S - E . n It is clear that u - uniform ...
Page 314
... uniformly for λ € VK . It is then clear that μ = V - 1 ( 2 ) is a non - nega- tive element of ba ( S , Σ ) such that lim 2 ( E ) = 0 uniformly for λ € K . ( E ) -0 To prove the converse , suppose there exists a non - negative ue ba ( S ...
... uniformly for λ € VK . It is then clear that μ = V - 1 ( 2 ) is a non - nega- tive element of ba ( S , Σ ) such that lim 2 ( E ) = 0 uniformly for λ € K . ( E ) -0 To prove the converse , suppose there exists a non - negative ue ba ( S ...
Page 360
... uniformly for a in some neighborhood of p . Show that if ső E „ ( x , z ) dz | ≤ M , then the convergence of Sf for a given c.o.n. system is localized if and only if max E , ( x , y ) | ≤M , < ∞ for each > 0 . 8 22 Suppose that ( Snf ) ...
... uniformly for a in some neighborhood of p . Show that if ső E „ ( x , z ) dz | ≤ M , then the convergence of Sf for a given c.o.n. system is localized if and only if max E , ( x , y ) | ≤M , < ∞ for each > 0 . 8 22 Suppose that ( Snf ) ...
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