Linear Operators: General theory |
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Page 186
... space ( S , 2 , μ ) constructed in Theorem 2 is called the product measure space of ... positive measure spaces ( S1 , 21 , μ1 ) and ( S2 , 2 , μ2 ) . For each E in ... measure spaces ( Si , 186 III.11.3 III . INTEGRATION AND SET FUNCTIONS.
... space ( S , 2 , μ ) constructed in Theorem 2 is called the product measure space of ... positive measure spaces ( S1 , 21 , μ1 ) and ( S2 , 2 , μ2 ) . 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 . ta PROOF . It is evidently sufficient to show that if { f } is a set of functions in L1 such that 0 ≤fa≤ go for ...
... positive measure space . Then the real partially ordered space L , ( S , Σ , μ ) , 1 ≤ p < ∞ , is a complete lattice . ta PROOF . It is evidently sufficient to show that if { f } is a set of functions in L1 such that 0 ≤fa≤ go for ...
Page 725
... measure . ( Hint . Consider the map ƒ ( s ) → ZA ( S ) f ( ps ) for each Ae 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 ƒ ( s ) → ZA ( S ) f ( ps ) for each Ae 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
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ