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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 Σ with v ( u , E ) < ∞ , the product Xf of ƒ with the characteristic function E 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 Σ with v ( u , E ) < ∞ , the product Xf of ƒ with the characteristic function E of E is totally measurable , the ...
Page 107
... ( u , ( A。~ A „ ) ' ) < 2ɛ . This shows that the sequence { f } of u - simple functions converges in u - measure to ßf . The fact that f ( ) is totally u - measurable follows from Lemma 8 . n Now let g be a continuous function defined on ...
... ( u , ( A。~ A „ ) ' ) < 2ɛ . This shows that the sequence { f } of u - simple functions converges in u - measure to ßf . The fact that f ( ) is totally u - measurable follows from Lemma 8 . n Now let g be a continuous function defined on ...
Page 178
... measurable , it will suffice then to show that ZF , 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 ZF , 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
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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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 Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field 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 topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ