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 u is additive , μ 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 ( u ) is defined so as to be equal to u if u itself is non - negative , to be additive if u is additive , μ 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 ) ≥ u ( 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 ) ≥ u ( 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 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
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ