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Page 177
Nelson Dunford, Jacob T. Schwartz. may be assumed that λ is real valued . A real valued set function can be represented as the difference of its positive and negative variations ( 4.11 ) and so we may also assume that 2 is positive . Let ...
Nelson Dunford, Jacob T. Schwartz. may be assumed that λ is real valued . A real valued set function can be represented as the difference of its positive and negative variations ( 4.11 ) and so we may also assume that 2 is positive . Let ...
Page 178
... assume that v ( μ , F ) < ∞ and v ( 2 , F ) < ∞ . n Since g is λ - measurable there is a sequence { g } of simple functions converging to g ( s ) for every s in F except on a set ECF with v ( 2 , E ) = 0 ( by Corollary 6.13 ( a ) ...
... assume that v ( μ , F ) < ∞ and v ( 2 , F ) < ∞ . n Since g is λ - measurable there is a sequence { g } of simple functions converging to g ( s ) for every s in F except on a set ECF with v ( 2 , E ) = 0 ( by Corollary 6.13 ( a ) ...
Page 675
... assume that f≥ 0. Let a ' be the com- plement of the set a = { sf ( s ) 1 } . Since fe L , it follows that μ ( a ) ... assumed that ƒ is in L1 . In view of Lemma 4 it may also be assumed that T is positive . Let e∞ = { s sup A ( T , k ) ...
... assume that f≥ 0. Let a ' be the com- plement of the set a = { sf ( s ) 1 } . Since fe L , it follows that μ ( a ) ... assumed that ƒ is in L1 . In view of Lemma 4 it may also be assumed that T is positive . Let e∞ = { s sup A ( T , k ) ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ