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Monotone convergence - help required

  1. Jul 14, 2012 #1
    Hi all,

    http://www.scribd.com/doc/100079521/Document-1 [Broken]

    Actually, I am trying to learn monotone convergence theorem, and I am stuck at one specific point, on the first page it says that ∫-∞→∞ f_n(x)dx = 1 for every n but the almost everywhere limit function is identically zero, what does it mean? how come the first is equal to 1 and the other is equal to zero??????

    Thanks in advance.
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Jul 14, 2012 #2

    tiny-tim

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    hi woundedtiger4! :smile:
    i don't understand why you're asking :confused:

    the limit (as n -> ∞) is obviously 0 (everywhere except x = 0)

    (and the integral happens to be 1, for all n, though that's less easy to prove)
     
  4. Jul 14, 2012 #3
    Draw yourself a picture. It might help you see what's going on. You have a sequence of normal densities with decreasing variance. For each n, the density must integrate to 1 because it's a normal density. But as n increases, the density gets narrower (by virtue of decreasing variance) and taller. So the sequence of densities converges to 0 everywhere except at x=0 where it's getting taller with increasing n. This should all be clear algebraically but sometimes a picture helps to clarify the concept.
     
  5. Jul 15, 2012 #4
    thanks a tonne
     
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