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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 μ , 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 μ , 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 μ∞ and Ма 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 ...
... measure zero ; the measure μ is not countably additive . In fact , using the relation between μ∞ and Ма 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 ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
Copyright | |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ