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Page 95
... o . A positive set function is a real valued or extended real valued set function which has no negative values . 2 DEFINITION . A set function μ defined on a 95 Integration and Set Functions 35 Finitely Additive Set Functions.
... o . A positive set function is a real valued or extended real valued set function which has no negative values . 2 DEFINITION . A set function μ defined on a 95 Integration and Set Functions 35 Finitely Additive Set Functions.
Page 182
... function f of Theorems 2 and 7 is called the Radon - Nikodým derivative of 2 with respect to u and is often denoted by da / du . Thus da / du is defined ... function μ2 on E2 the equation μ1 ( þ − 1 ( E ) ) = μ1⁄2 ( E ) defines an additive ...
... function f of Theorems 2 and 7 is called the Radon - Nikodým derivative of 2 with respect to u and is often denoted by da / du . Thus da / du is defined ... function μ2 on E2 the equation μ1 ( þ − 1 ( E ) ) = μ1⁄2 ( E ) defines an additive ...
Page 520
... functions defined on a convex set CC E. Then the function M defined by is a convex function . M ( x ) = sup M2 ( x ) π PROOF . For each л we have M ( au + ( 1 - x ) v ) ≤ aM2 ( u ) + ( 1 - x ) M , ( v ) , whenever u , v e C , and 0 ...
... functions defined on a convex set CC E. Then the function M defined by is a convex function . M ( x ) = sup M2 ( x ) π PROOF . For each л we have M ( au + ( 1 - x ) v ) ≤ aM2 ( u ) + ( 1 - x ) M , ( v ) , whenever u , v e C , and 0 ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite 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 TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ