Linear Operators: General theory |
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Page 108
... u - integrable if it differs by a null function from a function of the form n f = Σ XiXE¡ ' · . i = 1 where Ef1 ( x¿ ) , i = 1 , . , n , are disjoint sets in Σ with union S and where x = 0 if v ( u , E ; ) = ∞o . The phrases " u - ...
... u - integrable if it differs by a null function from a function of the form n f = Σ XiXE¡ ' · . i = 1 where Ef1 ( x¿ ) , i = 1 , . , n , are disjoint sets in Σ with union S and where x = 0 if v ( u , E ; ) = ∞o . The phrases " u - ...
Page 113
... integrable if and only if it is v ( μ ) -integrable . If f is μ - integrable ... ( u ) -measure , this lemma fol- lows directly from Lemma 8 and Definition 17 ... integral is linear on L ( III.2.18 113 INTEGRATION.
... integrable if and only if it is v ( μ ) -integrable . If f is μ - integrable ... ( u ) -measure , this lemma fol- lows directly from Lemma 8 and Definition 17 ... integral is linear on L ( III.2.18 113 INTEGRATION.
Page 180
... integrable . By Lemma 3 , fg is measur- able . The u - integrability of fg follows from Theorem 2.22 when it is observed ( by Theorem 4 ) that f ( ) g ( ) is μ - integrable . Theorem 4 applies in the same manner to prove the 2 - ...
... integrable . By Lemma 3 , fg is measur- able . The u - integrability of fg follows from Theorem 2.22 when it is observed ( by Theorem 4 ) that f ( ) g ( ) is μ - integrable . Theorem 4 applies in the same manner to prove the 2 - ...
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