Linear Operators: General theory |
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Page 106
... measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in E with v ( u , E ) < ∞ , the product XEf off with the characteristic function ZE of E is totally measurable , the ...
... measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in E with v ( u , E ) < ∞ , the product XEf off with the characteristic function ZE of E is totally measurable , the ...
Page 119
... u - measurable . Next suppose that we consider a function f ( vector or extended real - valued ) which is defined only on the complement of a u - null set NCS . Then we say that ƒ is u - measurable if the function g defined by g ( s ) ...
... u - measurable . Next suppose that we consider a function f ( vector or extended real - valued ) which is defined only on the complement of a u - null set NCS . Then we say that ƒ is u - measurable if the function g defined by g ( s ) ...
Page 178
... measurable , it will suffice then to show that fg is measurable . Thus · XF n ' we may and shall assume that v ( u , F ) < ∞ and v ( 2 , F ) < ∞ . Since g is 2 - measurable there is a sequence { g } of simple functions converging to g ...
... measurable , it will suffice then to show that fg is measurable . Thus · XF n ' we may and shall assume that v ( u , F ) < ∞ and v ( 2 , F ) < ∞ . Since g is 2 - measurable there is a sequence { g } of simple functions converging to g ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ