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 x iX Ei i = 1 where E , f ( x¿ ) , i = 1 , . . . , n , are disjoint sets in Σ with union S and where x = 0 if v ( u , E ; ) = ∞ . The phrases " u - ...
... u - integrable if it differs by a null function from a function of the form = n f = Σ x iX Ei i = 1 where E , f ( x¿ ) , i = 1 , . . . , n , are disjoint sets in Σ with union S and where x = 0 if v ( u , E ; ) = ∞ . The phrases " u - ...
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 u - integrable . Theorem 4 applies in the same manner to prove the λ - ...
... 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 u - integrable . Theorem 4 applies in the same manner to prove the λ - ...
Page 323
... u ( ds ) ≤ { u - ess sup | f ( s ) | } || μ || ( E ) , if f is u - simple . SEE 7 DEFINITION . A scalar valued measurable function ƒ is said to be integrable if there exists a sequence { f } of simple functions such that ( i ) fn ( s ) ...
... u ( ds ) ≤ { u - ess sup | f ( s ) | } || μ || ( E ) , if f is u - simple . SEE 7 DEFINITION . A scalar valued measurable function ƒ is said to be integrable if there exists a sequence { f } of simple functions such that ( i ) fn ( s ) ...
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ