Linear Operators: General theory |
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Page 106
... measurable on S , or , if u is understood , totally measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in with ( u , E ) < ∞ , the product Zef of f with the characteristic ...
... measurable on S , or , if u is understood , totally measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in with ( u , E ) < ∞ , the product Zef of f with the characteristic ...
Page 107
... ( u , ( A。○ A „ ) ' ) < 2ɛ . This shows that the sequence { nfn } of 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 ...
... ( u , ( A。○ A „ ) ' ) < 2ɛ . This shows that the sequence { nfn } of 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 ...
Page 178
... measurable . Conversely , let fg be u - measurable , and let ( S , E * , u ) and ( S , E * , 2 ) be the Lebesgue extensions of ( S , E , u ) and ( S , E , 2 ) respec- tively . Then * * . We note first that a given function is measur ...
... measurable . Conversely , let fg be u - measurable , and let ( S , E * , u ) and ( S , E * , 2 ) be the Lebesgue extensions of ( S , E , u ) and ( S , E , 2 ) respec- tively . Then * * . We note first that a given function is measur ...
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ