Linear Operators: General theory |
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Page 143
... Borel measure ) of the o - field of Borel sets in [ a , b ] . Similarly , the Lebesgue - Stieltjes measure determined by a func- tion f of bounded variation on a finite or infinite interval is the Le- besgue extension of the Borel ...
... Borel measure ) of the o - field of Borel sets in [ a , b ] . Similarly , the Lebesgue - Stieltjes measure determined by a func- tion f of bounded variation on a finite or infinite interval is the Le- besgue extension of the Borel ...
Page 634
... Borel set if E is a Borel set . Now let E be a Borel set of measure zero . By Fubini's theorem ( III.11.9 ) , we have • + ∞ + ∞0 [ ** S ** % , ( E ) ( s , t ) ds dt = [ ** S *** ZE ( 8—1 ) dsdt Sto -∞ -∞ + ∞ = -∞ | | XE ( 81 ) ds ...
... Borel set if E is a Borel set . Now let E be a Borel set of measure zero . By Fubini's theorem ( III.11.9 ) , we have • + ∞ + ∞0 [ ** S ** % , ( E ) ( s , t ) ds dt = [ ** S *** ZE ( 8—1 ) dsdt Sto -∞ -∞ + ∞ = -∞ | | XE ( 81 ) ds ...
Page 838
... Borel measure ( or Borel - Lebesgue measure ) , construction of , ( 139 ) III.13.8 ( 223 ) Borel - Stieltjes measure , ( 142 ) Bound , of an operator , II.3.5 ( 60 ) in a partially ordered set , I.2.3 ( 4 ) in the ( extended ) real ...
... Borel measure ( or Borel - Lebesgue measure ) , construction of , ( 139 ) III.13.8 ( 223 ) Borel - Stieltjes measure , ( 142 ) Bound , of an operator , II.3.5 ( 60 ) in a partially ordered set , I.2.3 ( 4 ) in the ( extended ) real ...
Contents
Preliminary Concepts | 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 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ