Linear Operators: General theory |
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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 lim μ ( C ) So \ # ( 9 ) −1 ( p ) \ μu ( dq ) is called the Lebesgue set of the function f . = = 0 Clearly , the Lebesgue ...
... Lebesgue integrable function defined on an open set in Euclidean n - space . The set of all points p at which 1 lim μ ( C ) So \ # ( 9 ) −1 ( p ) \ μu ( dq ) is called the Lebesgue set of the function f . = = 0 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
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ