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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 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 184
... function defined on a field Σ ; of subsets of a set S. Then there is a unique additive set function μ defined on the smallest field Σ in SX ... XS containing & such that μ ( E ) = II ... set 184 III.11.1 III . INTEGRATION AND SET FUNCTIONS.
... function defined on a field Σ ; of subsets of a set S. Then there is a unique additive set function μ defined on the smallest field Σ in SX ... XS containing & such that μ ( E ) = II ... set 184 III.11.1 III . INTEGRATION AND SET FUNCTIONS.
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