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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 , E , u ) 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 , E , u ) is a finite measure space , an ...
Page 722
... everywhere for fe Lp , 1 ≤p < ∞ . Show that if S is a topological space , and if , for each λ > 0 , R ( 2 .; A ) f ... everywhere . 1 22 Let T be a transformation in L1 , T≥ 0 , T1 ≤ 1 , and A ( n ) = A ( T , n ) . Show that if ƒ is ...
... everywhere for fe Lp , 1 ≤p < ∞ . Show that if S is a topological space , and if , for each λ > 0 , R ( 2 .; A ) f ... everywhere . 1 22 Let T be a transformation in L1 , T≥ 0 , T1 ≤ 1 , and A ( n ) = A ( T , n ) . Show that if ƒ is ...
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