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Page 179
... ΕΕΣ . Let g be a non - negative 2 - measurable function defined on S. Then fg is u - measurable , and √ ̧g ( s ) λ ( ds ) = √¿ † ( s ) g ( s ) μ ( ds ) , ΕΕΣ . PROOF . The μ - measurability of fg follows from Lemma 3. If we let H be ...
... ΕΕΣ . Let g be a non - negative 2 - measurable function defined on S. Then fg is u - measurable , and √ ̧g ( s ) λ ( ds ) = √¿ † ( s ) g ( s ) μ ( ds ) , ΕΕΣ . PROOF . The μ - measurability of fg follows from Lemma 3. If we let H be ...
Page 180
... ΕΕΣ . Then a function g on S to the B - space X is 2 - integrable if and only if fg is u - integrable , and in this case we have [ * ] √g_g ( s ) 2 ( ds ) = { f ( s ) g ( s ) μ ( ds ) , E ΕΕΣ . PROOF . Let g be 2 - integrable so that ...
... ΕΕΣ . Then a function g on S to the B - space X is 2 - integrable if and only if fg is u - integrable , and in this case we have [ * ] √g_g ( s ) 2 ( ds ) = { f ( s ) g ( s ) μ ( ds ) , E ΕΕΣ . PROOF . Let g be 2 - integrable so that ...
Page 181
... ΕΕΣ . PROOF . Using the Radon - Nikodým theorem ( Theorem 2 ) we find ( u ) -integrable functions g and h such that Since it follows that λ ( E ) = √g ( s ) v ( μ , ds ) , ΕΕΣ , μ ( E ) = [ h ( s ) v ( u , ds ) , E ΕΕΣ . v ( u , E ) ...
... ΕΕΣ . PROOF . Using the Radon - Nikodým theorem ( Theorem 2 ) we find ( u ) -integrable functions g and h such that Since it follows that λ ( E ) = √g ( s ) v ( μ , ds ) , ΕΕΣ , μ ( E ) = [ h ( s ) v ( u , ds ) , E ΕΕΣ . v ( u , E ) ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ