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Page 106
... measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in Σ with v ( u , E ) < ∞ , the product Zef of ƒ with the characteristic function ΧΕ of E is totally measurable , the ...
... measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in Σ with v ( u , E ) < ∞ , the product Zef of ƒ with the characteristic function ΧΕ of E is totally measurable , the ...
Page 107
... ( u , ( A。 An ) ' ) < 2ɛ . This shows that the sequence { Bnfn } 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 defined ...
... ( u , ( A。 An ) ' ) < 2ɛ . This shows that the sequence { Bnfn } 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 defined ...
Page 178
... ( u , ds ) E 0 for s in E except in a set A with v ( μ , A ) = 0. Thus gn ( s ) f ( s ) g ( s ) f ( s ) , 8 F - A , and Corollary 6.14 shows that XFfg is μ - measurable . Conversely , let fg be u - measurable , and let ( S , Σ * , μ ) and ...
... ( u , ds ) E 0 for s in E except in a set A with v ( μ , A ) = 0. Thus gn ( s ) f ( s ) g ( s ) f ( s ) , 8 F - A , and Corollary 6.14 shows that XFfg is μ - measurable . Conversely , let fg be u - measurable , and let ( S , Σ * , μ ) and ...
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