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Page 143
... 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 measurable sets in [ a , b ] are the sets in the Lebesgue extension ( relative to Borel measure ) ...
... 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 measurable sets in [ a , b ] are the sets in the Lebesgue extension ( relative to Borel measure ) ...
Page 218
... Lebesgue integrable function defined on an open set in Euclidean n - space . The set of all points p at which 1 limo μ ( C ) Sc ( f ( 9 ) − f ( p ) | μ ( dq ) = 0 is called the Lebesgue set of the function f . - Clearly , the Lebesgue ...
... Lebesgue integrable function defined on an open set in Euclidean n - space . The set of all points p at which 1 limo μ ( C ) Sc ( f ( 9 ) − f ( p ) | μ ( dq ) = 0 is called the Lebesgue set of the function f . - Clearly , the Lebesgue ...
Page 223
... Lebesgue - Stieltjes integral I = Sag ( s ) dh ( s ) exists . Let ƒ be a continuous increasing function on an open interval ( c , d ) , with f ( c ) = a , f ( d ) b . Show that the Lebesgue - Stieltjes integral = exists and is equal to ...
... Lebesgue - Stieltjes integral I = Sag ( s ) dh ( s ) exists . Let ƒ be a continuous increasing function on an open interval ( c , d ) , with f ( c ) = a , f ( d ) b . Show that the Lebesgue - Stieltjes integral = exists and is equal to ...
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