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 limo μ ( C ) √ ( 1 ( 9 ) − 1 ( 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 ) √ ( 1 ( 9 ) − 1 ( p ) | μ ( dq ) = 0 is called the Lebesgue set of the function f . Clearly , the Lebesgue ...
Page 223
... Lebesgue - Stieltjes integral I fag ( 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 I ...
... Lebesgue - Stieltjes integral I fag ( 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 I ...
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ