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Page 104
... of equivalence classes is well defined . μ , It is customary to speak of the elements of F ( S , Σ , μ , X ) as if they were functions rather than sets of equivalent functions and this we shall ordinarily do . Thus , we shall write f ...
... of equivalence classes is well defined . μ , It is customary to speak of the elements of F ( S , Σ , μ , X ) as if they were functions rather than sets of equivalent functions and this we shall ordinarily do . Thus , we shall write f ...
Page 196
... F is a u - measurable function whose values are in L „ ( T , Σr , λ ) , 1 ≤ p < ∞ . For each s in S , F ( s ) is an equivalence class of functions , any pair of whose members coincide 2 - almost every- where . If for each s we select ...
... F is a u - measurable function whose values are in L „ ( T , Σr , λ ) , 1 ≤ p < ∞ . For each s in S , F ( s ) is an equivalence class of functions , any pair of whose members coincide 2 - almost every- where . If for each s we select ...
Page 199
... function of t , is equal to the element Ss F ( s ) u ( ds ) of L „ ( T , ΣT , λ , X ) . ' T PROOF . Let Σ be partitioned into a sequence { E } of disjoint sets of finite 2 - measure . For 1 ≤ p ≤ ∞ let L = L , ( T , Σr , λ , X ) and ...
... function of t , is equal to the element Ss F ( s ) u ( ds ) of L „ ( T , ΣT , λ , X ) . ' T PROOF . Let Σ be partitioned into a sequence { E } of disjoint sets of finite 2 - measure . For 1 ≤ p ≤ ∞ let L = L , ( T , Σr , λ , X ) and ...
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