Linear Operators: General theory |
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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 97
... additive set function u is important because it dominates μ in the sense that v ( μ , E ) ≥ │μ ( E ) | for E e 2 ; the reader should test his com- prehension of Definition 4 below by proving that v ( μ ) is the smallest of the non ...
... additive set function u is important because it dominates μ in the sense that v ( μ , E ) ≥ │μ ( E ) | for E e 2 ; the reader should test his com- prehension of Definition 4 below by proving that v ( μ ) is the smallest of the non ...
Page 137
... additive set functions on E. The desired result follows from Theorem 8. Q.E.D. The next results on the extension of measures make use of inter- esting relations between the topology of a space and certain measures which may be defined ...
... additive set functions on E. The desired result follows from Theorem 8. Q.E.D. The next results on the extension of measures make use of inter- esting relations between the topology of a space and certain measures which may be defined ...
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