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Page 100
... everywhere , or , if μ 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 μ 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 724
... everywhere for every feL ( S , 2 , m ) . 33 " n Zj = 1 ( Dunford - Miller ) Let S , 2 , m , q , be as in Exercise 31. Show that lim∞ n - 1f ( p's ) converges in the mean of L1 ( S , Σ , m ) for every fe L1 ( S , Z , m ) only if there ...
... everywhere for every feL ( S , 2 , m ) . 33 " n Zj = 1 ( Dunford - Miller ) Let S , 2 , m , q , be as in Exercise 31. Show that lim∞ n - 1f ( p's ) converges in the mean of L1 ( S , Σ , m ) for every fe L1 ( S , Z , m ) only if there ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ