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Total variation of f on [0,1]

  1. Aug 17, 2007 #1
    1. The problem statement, all variables and given/known data
    Let [itex]f_n, f: [0,1]\rightarrow \mathbb{R}[/itex], and [itex]f_n(x)\rightarrow f(x)[/itex] for each [itex]x \in [0,1][/itex].

    I need to show the following two things:
    a. [itex]T_0^1(f)\leq \lim\inf_{n\rightarrow \infty} T_0^1(f_n)[/itex], and

    b. if each [itex]f_n[/itex] is absolutely continuous and [itex]T_0^1(f_n)\leq 1[/itex] for each [itex]n[/itex], then [itex]T_0^1(f) = \lim_{n\rightarrow \infty} T_0^1(f_n)[/itex].



    2. Relevant equations

    We denote [itex]T_0^1(f)[/itex] as the total variation of f on [itex][0,1][/itex].




    3. The attempt at a solution

    a.
    If [itex]f_n[/itex] is not of bounded variation, then [itex]T_0^1(f_n)=\infty[/itex] and we're done.

    So assume [itex]f_n[/itex] is of bounded variation. Then since [itex]f_n(x) \rightarrow f(x)[/itex] for each x, then for a partition [itex]0=t_0 < t_1 < ... < t_N= 1 [/itex],

    [itex] |f_n(t_j)-f_n(t_{j-1})| \rightarrow |f(t_j)-f(t_{j-1})| [/itex].

    So [itex] \sum_j|f_n(t_j)-f_n(t_{j-1})| \rightarrow \sum_j |f(t_j)-f(t_{j-1})| [/itex].
    Take the sup from both sides and so we have [itex]T_0^1(f_n)\rightarrow T_0^1(f)[/itex].

    I would like to use Fatou's at some point but I would like some hints on how to change the idea of total variation into a sequence of nonnegative measurable functions....
     
  2. jcsd
  3. Aug 17, 2007 #2

    EnumaElish

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    fn are bounded. Let L be a lower bound for all n.

    If L > 0, then all fn are positive.

    If L < 0, then you can define gn = fn - L > 0.

    If fn is (abs.) cont. then it is measurable (i.e. Borel).
     
    Last edited: Aug 17, 2007
  4. Aug 17, 2007 #3
    Definition of Total Variation: Let f(t) be complex-valued function defined on the interval [0,1]. Let [itex]P: 0=t_0 < t_1 < ... < t_N = 1 [/itex] be a partition of the unit interval. Then
    [itex]T_0^1(f) = \sup_{P} \:\sum_{i} |f(t_i)-f(t_i-1)| [/itex]
    where we take the supremum over all partitions of the unit interval.

    Fatou's Lemma: Let [itex]\{f_n\}_n [/itex] be a sequence of nonnegative measurable functions on [0,1]. Assume [itex]f_n(x)\rightarrow f(x)[/itex] for almost all x. Then
    [itex]\int f \leq \lim\inf_{n\rightarrow \infty} \int f_n[/itex].
     
    Last edited: Aug 17, 2007
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