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Page 137
... field B containing all the closed sets of a given topological space S is called the Borel field of S , and the sets ... G in Σ whose interior contains E such that │μ ( C ) ] < ɛ for every C in with CCG - F . For a complex or extended real ...
... field B containing all the closed sets of a given topological space S is called the Borel field of S , and the sets ... G in Σ whose interior contains E such that │μ ( C ) ] < ɛ for every C in with CCG - F . For a complex or extended real ...
Page 143
... field . Now if E1UN1 = EUN2 and N1C M1 , NCM , let M = M1UM2 so that EUM EUM and thus μ ( E1 ) = μ ( Е1 ○ M ) = μ ... g is u - integrable then the integral ( , g ( s ) μ ( ds ) is often written g ( s ) df ( s ) . In the case where f ( s ) ...
... field . Now if E1UN1 = EUN2 and N1C M1 , NCM , let M = M1UM2 so that EUM EUM and thus μ ( E1 ) = μ ( Е1 ○ M ) = μ ... g is u - integrable then the integral ( , g ( s ) μ ( ds ) is often written g ( s ) df ( s ) . In the case where f ( s ) ...
Page 168
... field , and clearly the smallest field containing . An elementary induction shows that C1 , n = 1 , 2 , . . . , is countable and thus Σ is a countable field . Q.E.D. 5 LEMMA . Let ( S , E , μ ) be a positive measure space and G a ...
... field , and clearly the smallest field containing . An elementary induction shows that C1 , n = 1 , 2 , . . . , is countable and thus Σ is a countable field . Q.E.D. 5 LEMMA . Let ( S , E , μ ) be a positive measure space and G a ...
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