Use Theorem 24.6 to show that the martingale of Example 25.14 is uniformly integrable. Data from example

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Use Theorem 24.6 to show that the martingale of Example 25.14 is uniformly integrable.

Data from example 25.14

Example 25.14 Consider in R" the half-open squares Qk (2):=z+[0,2 k)", kez, ze 2-kz", with lower left corner

where the supo ranges over all cubes Q- Qk (2) such that ke Z, z  2-kZ" and xEQ, and the submartingale

Data from theorem 24.6

Theorem 24.6 (convergence of UI submartingales) Let (Un)neN be a submartin- gale on the o-finite filtered

For the last estimate we use that on the set {u > Lw} {u { Lw} L0 This follows from the dominated convergence(ii)  (iii). Because of uniform integrability we have for some >0 and a suitable w E L (A) = [\u|d=] - S 1

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