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Homework Help: Uniform Convergence problem

  1. Dec 28, 2009 #1
    I'm trying to better understand convergence so I made upa problem for myself based on an example from class. I want to know if I'm answering my own questions correctly.

    Define a sequence of functions fn(x) = 1 if x is in {r1, r2, ... , rn} and 0 otherwise. Where r1, r2, ... , rn are the first n rational numbers in some enumeration of all rational numbers. fn converges pointwise to the dirichlet function. But, can we say anything else about how fn --> f?

    Uniform Convergence
    Given e > 0 is there an N such that when n > N |fn - f | < e for all x? No. Just let e=.5 we cal alays find an x value where |fn - f | = 1. That is, a rational number that has not yet been listed by the time we reach n.

    Convergence in Measure
    Yes. The measure of the set where f and fn are not the same is *always* 0.

    Almost Uniform Convergence
    Yes. If we let A, the set of measure less the any e where uniform convergence fails be Q, the rationals I think we have almost uniform convergence. since m{Q}=0 < e for all e > 0.

    Convergence in LP
    Yes, the Lp norm of the fn and f is always 0 anyway.
  2. jcsd
  3. Dec 28, 2009 #2
    Re: Convergence

    Yup. You've got it.
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