Linear Operators: General theory |
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Page 106
... measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in with v ( u , E ) < ∞ , the product Zef of f with the characteristic function ZE of E is totally measurable , the ...
... measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in with v ( u , E ) < ∞ , the product Zef of f with the characteristic function ZE of E is totally measurable , the ...
Page 107
... u - measurable . Let { ẞn } be a sequence of finitely valued u - simple functions with BB in u - measure . Given & > 0 ẞn ɛ there is , as before , a constant M and a set Д such that v * ( μ , A ' ) < ε and ẞ ( s ) ≤ M , se Ap . Let d ...
... u - measurable . Let { ẞn } be a sequence of finitely valued u - simple functions with BB in u - measure . Given & > 0 ẞn ɛ there is , as before , a constant M and a set Д such that v * ( μ , A ' ) < ε and ẞ ( s ) ≤ M , se Ap . Let d ...
Page 178
... measurable , it will suffice then to show that fg is measurable . Thus we may and shall assume that v ( u , F ) < ∞ and v ( 2 , F ) < ∞ . Since g is 2 - measurable there is a sequence { g } of simple functions converging to g ( s ) ...
... measurable , it will suffice then to show that fg is measurable . Thus we may and shall assume that v ( u , F ) < ∞ and v ( 2 , F ) < ∞ . Since g is 2 - measurable there is a sequence { g } of simple functions converging to g ( s ) ...
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