Linear Operators: General theory |
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Page 136
... field and a uniquely determined smallest o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all subsets of S , which contains a given family τ . The intersection of all fields ...
... field and a uniquely determined smallest o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all subsets of S , which contains a given family τ . The intersection of all fields ...
Page 166
... field , then Σ ( E ) is a σ - field . Σ ( Ε ) is called the restriction of E to E. If I is a field , Ε € Σ1 , and E is the o - field generated by 1 , then we can easily show that Σ ( E ) is the o - field generated by 1 ( E ) . Indeed ...
... field , then Σ ( E ) is a σ - field . Σ ( Ε ) is called the restriction of E to E. If I is a field , Ε € Σ1 , and E is the o - field generated by 1 , then we can easily show that Σ ( E ) is the o - field generated by 1 ( E ) . Indeed ...
Page 201
... field 2 , is the o - field E. An elementary set in S is one of the form Pa E where E is in E , and , for all but a finite number of a , ES . Equivalently , an elementary set in S is one having the form SXE for some л and some Е in & . The ...
... field 2 , is the o - field E. An elementary set in S is one of the form Pa E where E is in E , and , for all but a finite number of a , ES . Equivalently , an elementary set in S is one having the form SXE for some л and some Е in & . The ...
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