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Page 108
... integrable u - simple function " and " u - integrable simple function " will be used interchan- geably . For an Ee the integral over E of a u - integrable simple func- tion h is defined by the equation n S_h ( s ) μ ( ds ) = S_ f ( s ) u ...
... integrable u - simple function " and " u - integrable simple function " will be used interchan- geably . For an Ee the integral over E of a u - integrable simple func- tion h is defined by the equation n S_h ( s ) μ ( ds ) = S_ f ( s ) u ...
Page 113
... integrable if and only if it is v ( u ) -integrable . If f is u - integrable so is f ( · ) . If { f } is a sequence of u - integrable simple functions determining f in accordance with the preced- ing definition , then the sequence ...
... integrable if and only if it is v ( u ) -integrable . If f is u - integrable so is f ( · ) . If { f } is a sequence of u - integrable simple functions determining f in accordance with the preced- ing definition , then the sequence ...
Page 180
... u ( ds ) , E ΕΕΣ . PROOF . Suppose that g is 2 - integrable . By Lemma 3 , fg is measur- able . The u - integrability of fg follows from Theorem 2.22 when it is observed ( by Theorem 4 ) that f ( · ) g ( ) ] is u - integrable . Theorem ...
... u ( ds ) , E ΕΕΣ . PROOF . Suppose that g is 2 - integrable . By Lemma 3 , fg is measur- able . The u - integrability of fg follows from Theorem 2.22 when it is observed ( by Theorem 4 ) that f ( · ) g ( ) ] is u - integrable . Theorem ...
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