Linear Operators: General theory |
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Page 106
... measurable , the function f is said to be μ - measurable or , if u is understood , simply measurable . A set A is measurable if XA is measurable . Symbols for the set of measurable functions are M ( S , Σ , μ , X ) , M ( S , E , μ ) , M ...
... measurable , the function f is said to be μ - measurable or , if u is understood , simply measurable . A set A is measurable if XA is measurable . Symbols for the set of measurable functions are M ( S , Σ , μ , X ) , M ( S , E , μ ) , M ...
Page 107
... u - simple functions converges in u - measure to ßf . The fact that f ( ) is totally u - measurable follows from Lemma 8 . Now let g be a continuous function defined on the field of scalars and let ẞ be totally u - measurable . Let { f } ...
... u - simple functions converges in u - measure to ßf . The fact that f ( ) is totally u - measurable follows from Lemma 8 . Now let g be a continuous function defined on the field of scalars and let ẞ be totally u - measurable . Let { f } ...
Page 178
... measurable , it will suffice then to show that Xfg is measurable . Thus we may and shall assume that v ( u , F ) < ∞ and v ( 2 , F ) < ∞ . Since g is λ - measurable there is a sequence { g } of simple functions converging to g ( s ) ...
... measurable , it will suffice then to show that Xfg is measurable . Thus we may and shall assume that v ( u , F ) < ∞ and v ( 2 , F ) < ∞ . Since g is λ - measurable there is a sequence { g } of simple functions converging to g ( s ) ...
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