kkŽm 1. ž/ m 2 Ž.q For the proof of this proposition, it is sufficient to show that H xkkÝ 1 kŽ.j 22 j, k. Ž . k 1 ž/ j k k rewritten as H x kŽ.2 x H. k 1 k k 1. Then we obtain H xkkk2 1 Ý Ž.1 k 221 j Žk 1. j k 1 ½5ž/ j k 1 xkkž/ 2 Ý 1 kŽ.j 22 j j k 1 xkkž/Ý 1 kŽ.j 22 j, .
536 S. N. ETHIER AND T. G. KURTZ For m 1 and f BEŽ m.ŽŽ..Ž.², define F B PP E by F, : ff H m fd m, where denotes the mfold product measure of, and let E L mŽ.Ž Ž ..EB P be the collection of functions of this form. Note that L mŽ.Ž Ž ..Eis a subspace of B P closed under bounded pointwise conver gence. For f DŽBŽm.., define AF by f AF Ž. Ý Ž.² Žm. f,m .
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