Linear Operators: General theory |
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Page 143
... measure on as well as for its extension on * should cause no confusion . The measure of Lebesgue in an interval [ a , b ] of real numbers may be defined as the Lebesgue extension of the Borel measure in [ a , b ] . The Lebesgue ...
... measure on as well as for its extension on * should cause no confusion . The measure of Lebesgue in an interval [ a , b ] of real numbers may be defined as the Lebesgue extension of the Borel measure in [ a , b ] . The Lebesgue ...
Page 186
... measure with the desired properties . To prove the uniqueness of u , we note that it follows in the same way that ... measure space ( S , E , μ ) constructed in Theorem 2 is called the product measure space of the measure spaces ( Sn ...
... measure with the desired properties . To prove the uniqueness of u , we note that it follows in the same way that ... measure space ( S , E , μ ) constructed in Theorem 2 is called the product measure space of the measure spaces ( Sn ...
Page 405
... measure zero ; the measure μ is not countably additive . In fact , using the relation between and μ , it is easy to see that l is the union of a countable family of u - null sets . These relationships may be summed up in the following ...
... measure zero ; the measure μ is not countably additive . In fact , using the relation between and μ , it is easy to see that l is the union of a countable family of u - null sets . These relationships may be summed up in the following ...
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