Linear Operators: General theory |
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Page 95
... valued function and ƒ a vector ( or scalar ) valued function . Thus , even if the integration process is defined with respect to a scalar valued set function u , the resulting integral v may be a vector valued set function . It is ...
... valued function and ƒ a vector ( or scalar ) valued function . Thus , even if the integration process is defined with respect to a scalar valued set function u , the resulting integral v may be a vector valued set function . It is ...
Page 319
... valued measures is uniformly countably additive on 1 , which contradicts the assumption that yu ( E ) > ɛ , m = 1 , 2 , . . .. This proves the first assertion . If u is not 2 - continuous , then for ... VALUED MEASURES Vector Valued Measures.
... valued measures is uniformly countably additive on 1 , which contradicts the assumption that yu ( E ) > ɛ , m = 1 , 2 , . . .. This proves the first assertion . If u is not 2 - continuous , then for ... VALUED MEASURES Vector Valued Measures.
Page 323
... valued measurable function f is said to be integrable if there exists a sequence { f } of simple functions such that ( i ) fn ( s ) converges to f ( s ) u - almost everywhere ; ( ii ) the ... valued and u IV.10.7 323 VECTOR VALUED MEASURES.
... valued measurable function f is said to be integrable if there exists a sequence { f } of simple functions such that ( i ) fn ( s ) converges to f ( s ) u - almost everywhere ; ( ii ) the ... valued and u IV.10.7 323 VECTOR VALUED MEASURES.
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian 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