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Page 28
... exists a do e D , such that g ( f ( p ) , f ( q ) ) < ɛ if p≥ do , q≥ do . 5 LEMMA . If f is a generalized Cauchy sequence in a complete metric space X , there exists a pe X such that lim f ( d ) n = = p . .... PROOF . Let de D be ...
... exists a do e D , such that g ( f ( p ) , f ( q ) ) < ɛ if p≥ do , q≥ do . 5 LEMMA . If f is a generalized Cauchy sequence in a complete metric space X , there exists a pe X such that lim f ( d ) n = = p . .... PROOF . Let de D be ...
Page 292
... exists for E € 21. Then lim μn ( E ) exists for E € Σ . n - ∞ N∞ PROOF . Let be the family of all sets E in E for which lim ( EF ) exists for each Fe 21 , and let 23 be the family of all sets E ∞416 2 2 - 1 in 22 for which EF € Z2 ...
... exists for E € 21. Then lim μn ( E ) exists for E € Σ . n - ∞ N∞ PROOF . Let be the family of all sets E in E for which lim ( EF ) exists for each Fe 21 , and let 23 be the family of all sets E ∞416 2 2 - 1 in 22 for which EF € Z2 ...
Page 362
... exists a complex - valued regular measure μ in C * with an n = fern ( y ) μ ( dy ) if and only if Σx__ ∞ λmnan ‡ n ( x ) \ dx < K , for all m ≥ 1. Show that there exists an f in L1 with a , = ( y ) f ( y ) dy if and only if in ...
... exists a complex - valued regular measure μ in C * with an n = fern ( y ) μ ( dy ) if and only if Σx__ ∞ λmnan ‡ n ( x ) \ dx < K , for all m ≥ 1. Show that there exists an f in L1 with a , = ( y ) f ( y ) dy if and only if in ...
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 converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ