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Page 96
... variation of μ . The set function v ( μ ) is defined so as to be equal to u if u itself is non - negative , to be additive if u is additive , μ and to be bounded if u is bounded . The 96 III.1.2 III . INTEGRATION AND SET FUNCTIONS.
... variation of μ . The set function v ( μ ) is defined so as to be equal to u if u itself is non - negative , to be additive if u is additive , μ and to be bounded if u is bounded . The 96 III.1.2 III . INTEGRATION AND SET FUNCTIONS.
Page 97
... additive set function u is important because it dominates μ in the sense that v ( u , E ) ≥ │μ ( E ) for E e 2 ; the reader should test his com- prehension of Definition 4 below by proving that v ( u ) is the smallest of the non ...
... additive set function u is important because it dominates μ in the sense that v ( u , E ) ≥ │μ ( E ) for E e 2 ; the reader should test his com- prehension of Definition 4 below by proving that v ( u ) is the smallest of the non ...
Page 126
Nelson Dunford, Jacob T. Schwartz. 4. Countably Additive Set Functions The basis for the present section is a countably additive set func- tion defined on a σ - field of subsets of a set . In this case the results of the preceding ...
Nelson Dunford, Jacob T. Schwartz. 4. Countably Additive Set Functions The basis for the present section is a countably additive set func- tion defined on a σ - field of subsets of a set . In this case the results of the preceding ...
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