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Page 95
Nelson Dunford, Jacob T. Schwartz. CHAPTER III Integration and Set Functions 1. Finitely Additive Set Functions In contrast to the terms real function , complex function , etc ... set function u defined on a 95 Integration and Set Functions.
Nelson Dunford, Jacob T. Schwartz. CHAPTER III Integration and Set Functions 1. Finitely Additive Set Functions In contrast to the terms real function , complex function , etc ... set function u defined on a 95 Integration and Set Functions.
Page 96
... variation of μ . The set function v ( u ) is defined so as to be equal to u if u itself is non - negative , to be additive if μ is additive , μ Є and to be bounded if u is bounded . 96 III.1.2 III . INTEGRATION AND SET FUNCTIONS.
... variation of μ . The set function v ( u ) is defined so as to be equal to u if u itself is non - negative , to be additive if μ is additive , μ Є and to be bounded if u is bounded . 96 III.1.2 III . INTEGRATION AND SET FUNCTIONS.
Page 184
... set S. We shall denote by & the family of all sets in the Cartesian product S = SX ... XS , of the form E = E1X En , where E¡ € Z¿ . i i Mi ... 1 LEMMA . For i = 1 , ... , n , let μ , be a finitely additive complex valued set function ...
... set S. We shall denote by & the family of all sets in the Cartesian product S = SX ... XS , of the form E = E1X En , where E¡ € Z¿ . i i Mi ... 1 LEMMA . For i = 1 , ... , n , let μ , be a finitely additive complex valued set function ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz 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