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Page 106
... S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in Σ with v ( u , E ) < ∞ , the product % f of ƒ with the characteristic function XE of E is totally measurable , the function f is ...
... S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in Σ with v ( u , E ) < ∞ , the product % f of ƒ with the characteristic function XE of E is totally measurable , the function f is ...
Page 169
Nelson Dunford, Jacob T. Schwartz. 1 Show that ƒ e TM ( S , Σ , μ ) if and only if for each ɛ > 0 there exists a set E , e Σ and a finite collection of disjoint sets Д1 , ... , AneΣ such that 1U ... U A , E , v ( u , E ) < ɛ , and sup ...
Nelson Dunford, Jacob T. Schwartz. 1 Show that ƒ e TM ( S , Σ , μ ) if and only if for each ɛ > 0 there exists a set E , e Σ and a finite collection of disjoint sets Д1 , ... , AneΣ such that 1U ... U A , E , v ( u , E ) < ɛ , and sup ...
Page 329
Nelson Dunford, Jacob T. Schwartz. 11. The Space TM ( S , Σ , μ ) We shall be concerned here with a set S , a σ - field Σ of its subsets , and a scalar valued countably additive set function μ on Σ . The symbol TM ( S , E , u ) ... TM(S, 2,μ)
Nelson Dunford, Jacob T. Schwartz. 11. The Space TM ( S , Σ , μ ) We shall be concerned here with a set S , a σ - field Σ of its subsets , and a scalar valued countably additive set function μ on Σ . The symbol TM ( S , E , u ) ... TM(S, 2,μ)
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