Linear Operators: General theory |
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Page 96
... additive or finitely additive if μ ( $ ) = O and μ ( А2 U А „ . . . U An ) = μ ( 4 ) + μ ( 42 ) + ... + μ ( An ) , for every finite family { 41 , ... , An } of disjoint subsets of 7 whose union is in T. For an example of a finitely ...
... additive or finitely additive if μ ( $ ) = O and μ ( А2 U А „ . . . U An ) = μ ( 4 ) + μ ( 42 ) + ... + μ ( An ) , for every finite family { 41 , ... , An } of disjoint subsets of 7 whose union is in T. For an example of a finitely ...
Page 126
... additive set func- tion defined on a σ - field of subsets of a set . In this case the results of the preceding sections can be considerably extended . DEFINITION . Let u be a vector valued , complex valued , or extended real valued additive ...
... additive set func- tion defined on a σ - field of subsets of a set . In this case the results of the preceding sections can be considerably extended . DEFINITION . Let u be a vector valued , complex valued , or extended real valued additive ...
Page 233
... additive integrals . In Section IV.10 we will pre- sent a Lebesgue type theory of integration of a scalar function with respect to a countably additive vector valued measure . Instances in which both the function and the measure are ...
... additive integrals . In Section IV.10 we will pre- sent a Lebesgue type theory of integration of a scalar function with respect to a countably additive vector valued measure . Instances in which both the function and the measure are ...
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