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Page 106
... simple functions . If for every E in Σ with v ( u , E ) < ∞ , the product % f of ƒ with the characteristic function ... functions are M ( S , Σ , μ , X ) , M ( S , Σ , μ ) , M ( S ) and symbols besides TM ( S ) for the set of totally ...
... simple functions . If for every E in Σ with v ( u , E ) < ∞ , the product % f of ƒ with the characteristic function ... functions are M ( S , Σ , μ , X ) , M ( S , Σ , μ ) , M ( S ) and symbols besides TM ( S ) for the set of totally ...
Page 108
... simple function " and " u - integrable simple function " will be used interchan- geably . For an EeΣ the integral over E of a μ - integrable simple func- tion h is defined by the equation h ( s ) μ ( ds ) = [ f ( s ) μ ( ds ) St ( s ) μ ...
... simple function " and " u - integrable simple function " will be used interchan- geably . For an EeΣ the integral over E of a μ - integrable simple func- tion h is defined by the equation h ( s ) μ ( ds ) = [ f ( s ) μ ( ds ) St ( s ) μ ...
Page 322
... function f defined on S is u - simple if it is a finite linear combination of characteristic functions of sets in * ; this is evidently the case if and only if ƒ is 2 - simple . It follows from Corollaries III.6.13 and III.6.14 that ƒ ...
... function f defined on S is u - simple if it is a finite linear combination of characteristic functions of sets in * ; this is evidently the case if and only if ƒ is 2 - simple . It follows from Corollaries III.6.13 and III.6.14 that ƒ ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz 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