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Page 120
... fi - f2 \ p 0 if and only if fi - f2 is a null function . The inequality in part ( b ) is known as Minkowski's ... v ( μ , ds ) ≤ f ̧ { f ( 8 ) + \ f2 ( $ ) } } 3 v ( μ , ds ) -1 = √s \ f1 ( $ ) | { 120 III.3.3 III . INTEGRATION AND ...
... fi - f2 \ p 0 if and only if fi - f2 is a null function . The inequality in part ( b ) is known as Minkowski's ... v ( μ , ds ) ≤ f ̧ { f ( 8 ) + \ f2 ( $ ) } } 3 v ( μ , ds ) -1 = √s \ f1 ( $ ) | { 120 III.3.3 III . INTEGRATION AND ...
Page 122
... fi , fe L , 81 , 82 € Lo , and fi - f2 and g1 - g2 are μ - null functions , then fig1 - f2g2 is a μ - null function ... v ( μ , ds ) v ( μ , E ) → 0 E = 0 uniformly in n ; ( iii ) for each ɛ > 0 there is a set E , in Σ with v ( μ , E ...
... fi , fe L , 81 , 82 € Lo , and fi - f2 and g1 - g2 are μ - null functions , then fig1 - f2g2 is a μ - null function ... v ( μ , ds ) v ( μ , E ) → 0 E = 0 uniformly in n ; ( iii ) for each ɛ > 0 there is a set E , in Σ with v ( μ , E ...
Page 273
... V ... - Thus f ( u ) > F ( u ) -e for u e S. Since fs ,, ( t ) F ( t ) , we have fi ( t ) = F ( t ) , and hence there is a neighborhood V , of t with ft ( u ) < F ( u ) + ɛ , u € V¿ . Let V11 , ... , Vt cover S , and define t = t1 ...
... V ... - Thus f ( u ) > F ( u ) -e for u e S. Since fs ,, ( t ) F ( t ) , we have fi ( t ) = F ( t ) , and hence there is a neighborhood V , of t with ft ( u ) < F ( u ) + ɛ , u € V¿ . Let V11 , ... , Vt cover S , and define t = t1 ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
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 closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism 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 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ