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Page 95
... ando . A positive set function is a real valued or extended real valued set function which has no negative values . 95 2 DEFINITION . A set function μ defined on a Integration and Set Functions Finitely Additive Set Functions.
... ando . A positive set function is a real valued or extended real valued set function which has no negative values . 95 2 DEFINITION . A set function μ defined on a Integration and Set Functions 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 d2 / du . Thus da / du is defined ... function μ on E the equation u1 ( p - 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 d2 / du . Thus da / du is defined ... function μ on E the equation u1 ( p - 1 ( E ) ) = μ2 ( E ) defines an additive set ...
Page 520
... functions defined on a convex set CC Ek . 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 − a ) v ) ≤ aM „ ( u ) + ( 1 − α ) M „ ( v ) , whenever u , ve C ...
... functions defined on a convex set CC Ek . 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 − a ) v ) ≤ aM „ ( u ) + ( 1 − α ) M „ ( v ) , whenever u , ve C ...
Contents
Preliminary Concepts | 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 ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ