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Page 106
... measurable on S are the functions in the closure TM ( S ) in F ( S ) of the μ - simple functions . If for every E in Σ with v ( u , E ) < ∞ , the product XEf of f with the characteristic function XE of E is totally measurable , the ...
... measurable on S are the functions in the closure TM ( S ) in F ( S ) of the μ - simple functions . If for every E in Σ with v ( u , E ) < ∞ , the product XEf of f with the characteristic function XE of E is totally measurable , the ...
Page 119
... measurable . = Next suppose that we consider a function f ( vector or extended real - valued ) which is defined only ... functions we make no change in F ( S , Z , u , X ) , or in any of the theorems or lemmas of this section . Finally , ...
... measurable . = Next suppose that we consider a function f ( vector or extended real - valued ) which is defined only ... functions we make no change in F ( S , Z , u , X ) , or in any of the theorems or lemmas of this section . Finally , ...
Page 179
... measurable function defined on S and ¿ ( E ) = √2 † ( s ) μ ( ds ) , E ΕΕΣ . Let g be a non - negative λ ... functions h for which the equation [ h ( s ) 2 ( ds ) = √ 2f ( s ) h ( s ) μ ( ds ) , E ΕΕΣ , is valid , then H clearly contains ...
... measurable function defined on S and ¿ ( E ) = √2 † ( s ) μ ( ds ) , E ΕΕΣ . Let g be a non - negative λ ... functions h for which the equation [ h ( s ) 2 ( ds ) = √ 2f ( s ) h ( s ) μ ( ds ) , E ΕΕΣ , is valid , then H clearly contains ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ