From: hrubin@odds.stat.purdue.edu (Herman Rubin) Subject: Re: A.e. pointwise convergence not topology? Date: 8 Jan 2000 10:42:23 -0500 Newsgroups: sci.math Summary: [missing] In article <3876B256.27B8A8BC@post.rwth-aachen.de>, Volker W. Elling wrote: >> On the other hand, if you have a sequence of functions converging in >> measure but not a.e. to f, every sequence has a subsequence converging >> a.e. to f and yet the original sequence doesn't converge to >> f. I.e. a.e. convergence is not topological. >Well, but it is still possible that a.e. convergence corresponds to >a topology that does not satisfy the first countability axiom. This is >almost certainly necessary because the related pointwise convergence >corresponds to a non-first-axiom topology too. No. In any topology, if a net does not converge to a point, there is a subnet no subnet of which converges to that point. There are other subnets than subsequences, but this is enough. Suppose a sequence converges in measure to f. Then every subsequence has a subsequence converging almost everywhere to f. Thus, almost everywhere convergence leads, at least on a space of finite measure, to convergence in measure. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 ============================================================================== From: bruck@math.usc.edu (Ronald Bruck) Subject: Re: A.e. pointwise convergence not topology? Date: Sat, 08 Jan 2000 09:27:20 -0800 Newsgroups: sci.math In article <857lsv$e7k@odds.stat.purdue.edu>, hrubin@odds.stat.purdue.edu (Herman Rubin) wrote: :In article <3876B256.27B8A8BC@post.rwth-aachen.de>, :Volker W. Elling wrote: :>> On the other hand, if you have a sequence of functions converging in :>> measure but not a.e. to f, every sequence has a subsequence converging :>> a.e. to f and yet the original sequence doesn't converge to :>> f. I.e. a.e. convergence is not topological. : :>Well, but it is still possible that a.e. convergence corresponds to :>a topology that does not satisfy the first countability axiom. This is :>almost certainly necessary because the related pointwise convergence :>corresponds to a non-first-axiom topology too. : :No. In any topology, if a net does not converge to a point, :there is a subnet no subnet of which converges to that point. :There are other subnets than subsequences, but this is enough. : :Suppose a sequence converges in measure to f. Then every :subsequence has a subsequence converging almost everywhere :to f. Thus, almost everywhere convergence leads, at least :on a space of finite measure, to convergence in measure. There is a concept in the literature (and which I used in my own thesis in 1969, when I wasn't able to FIND it in the literature) of "compactness in the topology of convergence in measure". As it stood, the idea was a little too abstract to be very useful (it played a role in a couple of corollaries), and I was interested (at that time) in equivalent formulations. Because of the existence of a.e. convergent subsequences (assume the measure space is finite), I thought the notion should somehow be related to pointwise boundedness, in some a.e. sense (but that's obviously not enough). Does anybody know of a useful, or elegant, equivalent formulation? Some reasonably general sufficiency conditions? In special spaces, such as L^p? --Ron Bruck