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 318
... Valued Measures The theorems on spaces of set functions proved in the last section permit us to develop a more satisfactory theory of vector valued count- ably additive set functions ( briefly , vector valued measures ) than we were ...
... Valued Measures The theorems on spaces of set functions proved in the last section permit us to develop a more satisfactory theory of vector valued count- ably additive set functions ( briefly , vector valued measures ) than we were ...
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
Preliminary Concepts | 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 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ