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 μ , 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 μ , 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 ) > & , m = 1 , 2 , . . .. This proves the first assertion . If μ 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 ) > & , m = 1 , 2 , . . .. This proves the first assertion . If μ is not 2 - continuous , then for ... VALUED MEASURES Vector Valued Measures.
Page 323
... valued measurable function ƒ 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 ƒ 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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ