New PDF release: Bergman Spaces and Related Topics in Complex Analysis:

By Hadenmalm & Zhu Borichev

ISBN-10: 0821837125

ISBN-13: 9780821837122

ISBN-10: 1641965363

ISBN-13: 9781641965361

ISBN-10: 3519914794

ISBN-13: 9783519914792

ISBN-10: 5319966016

ISBN-13: 9785319966018

This quantity grew out of a convention in honor of Boris Korenblum at the celebration of his eightieth birthday, held in Barcelona, Spain, November 20-22, 2003. The booklet is of curiosity to researchers and graduate scholars operating within the idea of areas of analytic functionality, and, particularly, within the concept of Bergman areas. This booklet is copublished with Bar-Ilan college (Ramat-Gan, Israel)

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Additional resources for Bergman Spaces and Related Topics in Complex Analysis: Proceedings of a Conference in Honor of Boris Korenblum's 80th Birthday, November 20-23, 2003

Sample text

Then inf fk , sup fk , lim inf fk , and lim sup fk k≥1 k≥1 k→∞ k→∞ are also µ-measurable. Proof. 1. As noted above, f : X → [−∞, ∞] is µ-measurable if and only if f −1 [−∞, a] is µ-measurable for each a ∈ R. 2. Suppose f, g : X → R are µ-measurable, Then (f + g)−1 (−∞, a) = r,s rational r+s

BM1 in G1 satisfying (⋆) . 3. Now let M1 U2 := U − Bi , i=1 F2 := {B | B ⊆ U2 , diam B < δ} , and, as above, find finitely many disjoint balls BM1 +1 , . . , BM2 in F2 such that M2 M2 Ln U− Bi i=1 = Ln U2 − Bi i=M1 +1 ≤ θLn (U2 ) ≤ θ 2 Ln (U ). 4. Continue this process to obtain a countable collection of disjoint balls such that Mk Ln U− Bi i=1 ≤ θ k Ln (U ) (k = 1, . . ). Since θ k → 0, the theorem is proved if Ln (U ) < ∞. 5 Covering theorems 39 Should Ln (U ) = ∞, we apply the above construction to each of the open sets Um := {x ∈ U | m < |x| < m + 1} (m = 0, 1, .

Then lim inf fk dµ ≤ lim inf k→∞ ∞ k→∞ fk dµ. Proof. Take g := j=1 aj χAj to be a nonnegative simple function less than or equal to lim inf k→∞ fk . Suppose the µ-measurable sets {Aj }∞ j=1 are disjoint and aj > 0 for j = 1, . . 3 Integrals and limit theorems 27 Fix 0 < t < 1. Then ∞ Aj = Bj,k, k=1 where Bj,k := Aj ∩ {x | fl (x) > taj for all l ≥ k}. Note Aj ⊇ Bj,k+1 ⊇ Bj,k (k = 1, . . ). Thus fk dµ ≥ ∞ Aj j=1 fk dµ ≥ and so lim inf k→∞ fk dµ ≥ t ∞ j=1 ∞ Bj,k fk dµ ≥ t aj µ(Aj ) = t ∞ aj µ(Bj,k ); j=1 g dµ.

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Bergman Spaces and Related Topics in Complex Analysis: Proceedings of a Conference in Honor of Boris Korenblum's 80th Birthday, November 20-23, 2003 by Hadenmalm & Zhu Borichev


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