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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 μ - 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 μ - null set . The phrase " almost everywhere " is customarily abbreviated " a.e. " Thus , if lim ...
Page 150
... everywhere . 13 COROLLARY . ( a ) If ( S , E , 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 , E , 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 723
... everywhere . ( Hint . Use the preceding exercise . ) 26 Let ( S , E , μ ) , 9 , w , and T be as in Exercise 24. Let ge L1 , g ( s ) > 0 almost everywhere , and let ƒ be non - negative and μ - measur- able . Show that lim n14∞ A ( T , n ) ...
... everywhere . ( Hint . Use the preceding exercise . ) 26 Let ( S , E , μ ) , 9 , w , and T be as in Exercise 24. Let ge L1 , g ( s ) > 0 almost everywhere , and let ƒ be non - negative and μ - measur- able . Show that lim n14∞ A ( T , n ) ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ