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Page 120
... fi and f2 lies in L ( S , E , μ ) , and f1 + f2p ≤ - = ( c ) fi - f2 \ p 0 if and only if f1 - f2 is a null ... v ( μ , ds ) | √s { \ f1 ( $ ) | + \ f2 ( 8 ) | } ” v ( μ , ds ) = { s \ f1 ( $ ) | { 120 III.3.3 III . INTEGRATION AND SET ...
... fi and f2 lies in L ( S , E , μ ) , and f1 + f2p ≤ - = ( c ) fi - f2 \ p 0 if and only if f1 - f2 is a null ... v ( μ , ds ) | √s { \ f1 ( $ ) | + \ f2 ( 8 ) | } ” v ( μ , ds ) = { s \ f1 ( $ ) | { 120 III.3.3 III . INTEGRATION AND SET ...
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 ... v ( μ , 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 ... v ( μ , 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 273
... v and ^ . Next , we note that for an arbitrary F € C ( S ) and an arbitrary pair s , te S there is an fs , te with ... fi ( t ) F ( t ) , and hence there is a neighborhood V of t with = ft ( u ) < F ( u ) + ɛ , u € Vt . Let V , ... , Vt ...
... v and ^ . Next , we note that for an arbitrary F € C ( S ) and an arbitrary pair s , te S there is an fs , te with ... fi ( t ) F ( t ) , and hence there is a neighborhood V of t with = ft ( u ) < F ( u ) + ɛ , u € Vt . Let V , ... , Vt ...
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