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Page 100
... everywhere , or , if u is understood , simply almost everywhere , or for almost all s in S , if it is true except for those points s in a u - null set . The phrase " almost everywhere " is customarily abbreviated " a.e. " Thus , if lim ...
... everywhere , or , if u is understood , simply almost everywhere , or for almost all s in S , if it is true except for those points s in a u - null set . The phrase " almost everywhere " is customarily abbreviated " a.e. " Thus , if lim ...
Page 150
... everywhere . 13 COROLLARY . ( a ) If ( S , Σ , μ ) is a measure space , a sequence of measurable functions convergent in measure has a subsequence which converges almost everywhere . ( b ) If ( S , Σ , μ ) is a finite measure space , an ...
... everywhere . 13 COROLLARY . ( a ) If ( S , Σ , μ ) is a measure space , a sequence of measurable functions convergent in measure has a subsequence which converges almost everywhere . ( b ) If ( S , Σ , μ ) is a finite measure space , an ...
Page 722
... everywhere . 22 Let T be a transformation in L , T≥0 , T1 ≤ 1 , and A ( n ) = A ( T , n ) . Show that if f is real , fe L1 , and supo≤n A ( n ) f≥0 almost everywhere , and infos , A ( n ) f≤0 almost everywhere , then f = 0 . 23 ...
... everywhere . 22 Let T be a transformation in L , T≥0 , T1 ≤ 1 , and A ( n ) = A ( T , n ) . Show that if f is real , fe L1 , and supo≤n A ( n ) f≥0 almost everywhere , and infos , A ( n ) f≤0 almost everywhere , then f = 0 . 23 ...
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