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Page 95
... ∞ . A positive set function is a real valued or extended real valued set function which has no negative values . 2 DEFINITION . A set function u defined on a 95 Integration and Set Functions 35 Finitely Additive Set Functions.
... ∞ . A positive set function is a real valued or extended real valued set function which has no negative values . 2 DEFINITION . A set function u 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 dλ / du . Thus dλ / du is defined ... function μ on 22 the equation ( 4 - 1 ( E ) ) == μ2 ( E ) defines an additive set ...
... function f of Theorems 2 and 7 is called the Radon - Nikodým derivative of 2 with respect to u and is often denoted by dλ / du . Thus dλ / du is defined ... function μ on 22 the equation ( 4 - 1 ( E ) ) == μ2 ( E ) defines an additive set ...
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 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 disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ