(Egorovs Theorem). Show that, if μ is finite, then X n X implies that X n X...
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For an arbitrary e > 0 to be kept fixed throughout and k ³ 1 integer, use Theorem 4 in order to conclude that there exists Nk = (e, k) > 0 such that μ(Ae,k) < e/2k, k ³ 1, where Ae,k = (|Xn X| ³ 1/k). Thus, if Ae = Ae,k, then μ (Ae) £ e. Finally, show that Xn (w) X(w) uniformly in w à Ae.
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Finiteness of implies by Theorem 4 in this chapter that X n as n X if and only if ...View the full answer
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Related Book For
An Introduction to Measure Theoretic Probability
ISBN: 978-0128000427
2nd edition
Authors: George G. Roussas
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