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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 7. 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 7. The intersection of all fields ...
Page 166
... o in μ - measure , it follows from Definition 2.17 that | ƒ ( · ) | P is μ - integrable and thus that fe Lp ( u ) ... field of subsets of E , and that Σ ( E ) is the family of all sets AE , A € Σ , and that if Σ is a σ - field , then Σ ( E ) ...
... o in μ - measure , it follows from Definition 2.17 that | ƒ ( · ) | P is μ - integrable and thus that fe Lp ( u ) ... field of subsets of E , and that Σ ( E ) is the family of all sets AE , A € Σ , and that if Σ is a σ - field , then Σ ( E ) ...
Page 201
... field , is the o - field E. An elementary set in S is one of the form PA E where E is in 2 and , for all but a finite number of x , ES . Equivalently , an elementary set in S is one having the form SXE for some л and some E in & . The ...
... field , is the o - field E. An elementary set in S is one of the form PA E where E is in 2 and , for all but a finite number of x , ES . Equivalently , an elementary set in S is one having the form SXE for some л and some E in & . The ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ