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Page 186
... space ( S , Σ , μ ) constructed in Theorem 2 is called the product measure space of ... positive measure spaces ( S1 , E1 , μ1 ) and ( S2 , Z2 , 2 ) . For each E in ... measure spaces ( Si , 186 III.11.3 III . INTEGRATION AND SET FUNCTIONS.
... space ( S , Σ , μ ) constructed in Theorem 2 is called the product measure space of ... positive measure spaces ( S1 , E1 , μ1 ) and ( S2 , Z2 , 2 ) . For each E in ... 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 go ...
... 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 go ...
Page 725
... measure . ( Hint . Consider the map f ( s ) → XA ( s ) f ( ps ) for each A € Σ with μ ( A ) < ∞ . ) Ε 37 Let ( S , Σ , μ ) be a positive measure space ... space , and let ( S , Z , μ ) be a reg- ular finite measure space . Let o be a ...
... measure . ( Hint . Consider the map f ( s ) → XA ( s ) f ( ps ) for each A € Σ with μ ( A ) < ∞ . ) Ε 37 Let ( S , Σ , μ ) be a positive measure space ... space , and let ( S , Z , μ ) be a reg- ular finite measure space . Let o be a ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ