Linear Operators: General theory |
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Page 136
... o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all subsets ... finite on if S is the union of a sequence { E } of sets in such that v ( μ , En ) < ∞ , n 1 , 2 , .... A measure space ( ...
... o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all subsets ... finite on if S is the union of a sequence { E } of sets in such that v ( μ , En ) < ∞ , n 1 , 2 , .... A measure space ( ...
Page 198
... finite . Since f is o - measurable , there is a se- quence { f } of o - measurable simple functions converging in o - measure to f . In virtue of Lemma 8.3 and the remark before Theorem 2 we may suppose that each f is a finite linear ...
... finite . Since f is o - measurable , there is a se- quence { f } of o - measurable simple functions converging in o - measure to f . In virtue of Lemma 8.3 and the remark before Theorem 2 we may suppose that each f is a finite linear ...
Page 290
... o - finite , and let E , be an increasing sequence of measurable sets of finite measure whose union is S. Using the theorem for L1 ( E ) = L1 ( En Σ ( En ) , μ ) , we obtain a sequence 00 = { g } of functions in L such that gn∞ ≤ x ...
... o - finite , and let E , be an increasing sequence of measurable sets of finite measure whose union is S. Using the theorem for L1 ( E ) = L1 ( En Σ ( En ) , μ ) , we obtain a sequence 00 = { g } of functions in L such that gn∞ ≤ x ...
Contents
Preliminary Concepts | 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 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ