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Page 186
... space ( S , E , μ ) constructed in Theorem 2 is called the product measure space of the ... positive measure spaces ( $ 1 , 21 , μ1 ) and ( S2 , Z2 , μ2 ) . For each E ... measure spaces ( Si , 186 III.11.3 III . INTEGRATION AND SET FUNCTIONS.
... space ( S , E , μ ) constructed in Theorem 2 is called the product measure space of the ... positive measure spaces ( $ 1 , 21 , μ1 ) and ( S2 , Z2 , μ2 ) . For each E ... measure spaces ( Si , 186 III.11.3 III . INTEGRATION AND SET FUNCTIONS.
Page 302
... positive measure space . Then the real partially ordered space L ( S , Σ , μ ) , 1 ≤ p < ∞ , is a complete lattice . PROOF . It is evidently sufficient to show that if { f } is a set of functions in L1 such that 0 ≤fago for some goe ...
... positive measure space . Then the real partially ordered space L ( S , Σ , μ ) , 1 ≤ p < ∞ , is a complete lattice . PROOF . It is evidently sufficient to show that if { f } is a set of functions in L1 such that 0 ≤fago for some goe ...
Page 725
... measure . ( Hint . Consider the map f ( s ) → ZA ( s ) f ( ps ) for each A e Σ with μ ( 4 ) < ∞ . ) 37 Let ( S , E , μ ) be a positive measure space ... space , and let ( S , Σ , μ ) be a reg- ular finite measure space . Let p be a mapping ...
... measure . ( Hint . Consider the map f ( s ) → ZA ( s ) f ( ps ) for each A e Σ with μ ( 4 ) < ∞ . ) 37 Let ( S , E , μ ) be a positive measure space ... space , and let ( S , Σ , μ ) be a reg- ular finite measure space . Let p be a mapping ...
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