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Page 122
... fi , fe L , 81 , 82 € Lo , and fi - f2 and g1 - g2 are u - null functions , then fig1 - f2g2 is a μ - null function ... E ) → 0 E ( iii ) for each ɛ > 0 there is a set E in Σ with v ( μ , E ̧ ) < ∞ and such that √s_ £ 2 \ fn ...
... fi , fe L , 81 , 82 € Lo , and fi - f2 and g1 - g2 are u - null functions , then fig1 - f2g2 is a μ - null function ... E ) → 0 E ( iii ) for each ɛ > 0 there is a set E in Σ with v ( μ , E ̧ ) < ∞ and such that √s_ £ 2 \ fn ...
Page 305
... e L , ( S , Σ , μ ) , f ; ≥ 0 , j 1 , 2 , -181i , f2 = -182 ; be decompositions of ƒ1 and ƒ1⁄2 into fi positive functions . Then Σgii + Σg2 ; is a decomposition of ƒ1 + ƒ1⁄2 , and let f1 = and so To ( f1 + f2 ) ≤ To ( f1 ) + To ( f2 ) ...
... e L , ( S , Σ , μ ) , f ; ≥ 0 , j 1 , 2 , -181i , f2 = -182 ; be decompositions of ƒ1 and ƒ1⁄2 into fi positive functions . Then Σgii + Σg2 ; is a decomposition of ƒ1 + ƒ1⁄2 , and let f1 = and so To ( f1 + f2 ) ≤ To ( f1 ) + To ( f2 ) ...
Page 312
... ( E | F ) = t ( E ) Ut ( F ) , T ( EF ) = T ( E ) t ( F ) , and τ ( E ' ) = τ ( E ) ' for all E , F € Σ . n PROOF . As ... fi ( t ) = 0 for t in G ( cf. I.5.2 ) . As before , we select a simple function g 1 % , in B ( S , E ) where A ...
... ( E | F ) = t ( E ) Ut ( F ) , T ( EF ) = T ( E ) t ( F ) , and τ ( E ' ) = τ ( E ) ' for all E , F € Σ . n PROOF . As ... fi ( t ) = 0 for t in G ( cf. I.5.2 ) . As before , we select a simple function g 1 % , in B ( S , E ) where A ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
Copyright | |
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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