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 , Σ , 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 , Σ , μ ) , M ...
Page 107
... u - measurable . Let { n } be a sequence of finitely valued u - simple functions with BnB in u - measure . Given ε > 0 ɛ there is , as before , a constant M and a set Д such that ʊ * ( μ , A ' ) < ɛ and ẞ ( s ) ≤ M , se A. Let 8 be a ...
... u - measurable . Let { n } be a sequence of finitely valued u - simple functions with BnB in u - measure . Given ε > 0 ɛ there is , as before , a constant M and a set Д such that ʊ * ( μ , A ' ) < ɛ and ẞ ( s ) ≤ M , se A. Let 8 be a ...
Page 149
... u - measurable if and only if ( i ) for every measurable set F with v ( μ , F ) < ∞ , the function f is u - essentially separably valued on F , and ( ii ) for every linear functional x * in X * the scalar function x * f on S is u - ...
... u - measurable if and only if ( i ) for every measurable set F with v ( μ , F ) < ∞ , the function f is u - essentially separably valued on F , and ( ii ) for every linear functional x * in X * the scalar function x * f on S is u - ...
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