From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: pointwise convergence to a steady function... Date: 28 Oct 2000 15:49:07 -0400 Newsgroups: sci.math Summary: [missing] In article <39FAB664.D63D070A@t-online.de>, Harald Stepputtis wrote: :Hi out there! :I am stuck with a math question, I don't know how to solve: :Given a series of function which are pointwise converge to a steady :(and monoton growing) function. Now, according to my professor, it :follows, that the serie of functions converges not only pointwise :but also regular (I hope this is the right translation, of :convergence of lim sup). :I hope one of you guys can help me. :Cheers :Harald :------- This sounds very much like Dini's Theorem, except some assumptions are dropped or misquoted. Of course, there may be other theorems around which use similar concepts. So: Dini's Theorem: Let X be a topological space, and for every x in X, let the sequence {f_n(x)} converge to f(x), where (1) all f_n are continuous real-valued functions on X, (2) f is a continuous function on X, (3) for every x in X, the sequence {f_n(x)} is monotone non-decreasing, (4) X is compact. Then the sequence {f_n} converges to f uniformly on X. (Instead of non-decreasing, one may assume non-increasing at every x in X.) The proof uses the covering definition of compactness. And Example 7.12 in "Counterexamples in Analysis" by B.R. Gelbaum and J.M.H. Olmsted demonstrates that dropping any of the four assumptions allows for non-uniform convergence. In particular, X can be a closed bounded subset of a Euclidean space. Hope it helps, ZVK(Slavek). ============================================================================== From: ikastan@uranus.uucp (Ilias Kastanas) Subject: Re: pointwise convergence to a steady function... Date: 31 Oct 2000 10:53:19 GMT Newsgroups: sci.math In article <8tg0r3$6q2@mcmail.cis.mcmaster.ca>, Zdislav V. Kovarik wrote: @In article <39FB455F.FF8C43F8@t-online.de>, @Harald Stepputtis wrote: @:Thank you very much for your gender help. I must excuse my bad @:english and my mistranslation of german mathematic names. @:I try to prove the following probability theorem: @:Given a series of not necessary continous distribution-functiions @:which converges to a continous distribution function by distribution @:(that means pointwise) then the convergence is even uniformly (there @:is a special name for this in probability theory, but I don't want @:to risk to translate it ;-)). @:A distribution function is a limited (or bounded?) function which @:is only capable of values 0 to 1. It is a monoton increasing @:function. It is not necessary continous. But it must be "continous @:from the right side" and may have countable many "jumping-points". @:Any ideas? @ @So, this is not in the Dini Theorem family. @ @Better textbooks of probablilty theory contain theorems of this @kind, well-formulated and proved. Some of them are known under the @name Helly's Theorem(s) in the context of Riemann-Stieltjes @integration. A book would do you a much better service than this @forum. (Why re-invent the wheel?) @ @Cheers, ZVK(Slavek). Say we mix up the hypotheses: Dini says "monotone sequence of continuous functions"; Not-Dini says "sequence of monotone functions". What luck -- the conclusion is still correct; convergence to a continuous f on a compact set X implies uniform convergence. (Not-Dini in this case is Polya). The proof is easy; sketch, with X = [a,b]: There exist finite- ly many points a = x_0 < x_1 < ... < x_t = b such that f increases by less than epsilon from each x_i to the next. At each x_i there is a K_i such that |f_n(x_i) - f(x_i)| < epsilon for n > K_i. Take n > maximum of the K_i and apply obvious inequalities. So, back to the original question, we can assert convergence is uniform on compact subsets of R. No, we can't hope for uniform convergence on all of R. Slavek is right, we could also invoke a Helly-type theorem (an 800-pound gorilla in place of our 300-pound one). Helly Selection, for instance: every sequence of distribution functions has a weakly convergent subsequence -- which here amounts to: there is a (right continuous, nondecreasing) g such that for each point of continuity x of g the subsequence converges to g(x). With a bit of fiddling we would get information about the sequence sup{|f_n(y) - f(y)| : y <= x}. Ilias