Linear Operators: General theory |
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Page 144
... Borel set B. Let In = { s n < s < + n } , and put μ ( B ) = lim ( IB ) for every Borel set ( the limit exists as an extended positive real number , since { u ( IB ) } is an increasing sequence ) . To see that u is countably additive ...
... Borel set B. Let In = { s n < s < + n } , and put μ ( B ) = lim ( IB ) for every Borel set ( the limit exists as an extended positive real number , since { u ( IB ) } is an increasing sequence ) . To see that u is countably additive ...
Page 341
... set S , and let 1 be the o - field generated by E. Let μ be a non - negative finite countably additive measure ... Borel sets in S. Prove that ( i ) rca ( S ) is weakly complete . ( ii ) A sequence { u } in rca ( S ) is a weak Cauchy sequence ...
... set S , and let 1 be the o - field generated by E. Let μ be a non - negative finite countably additive measure ... Borel sets in S. Prove that ( i ) rca ( S ) is weakly complete . ( ii ) A sequence { u } in rca ( S ) is a weak Cauchy sequence ...
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 • + ∞ √ √ 100 % . ( E ) ( s , t ) ds dt -1 80 - ∞ + J + ∞ [ + € [ + ZE ( s − t ) dsdt -∞ 00 - -∞ ∞- ∞ + J c ...
... 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 • + ∞ √ √ 100 % . ( E ) ( s , t ) ds dt -1 80 - ∞ + J + ∞ [ + € [ + ZE ( s − t ) dsdt -∞ 00 - -∞ ∞- ∞ + J c ...
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