bham10246
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Homework Statement
Let f_n, f: [0,1]\rightarrow \mathbb{R}, and f_n(x)\rightarrow f(x) for each x \in [0,1].
I need to show the following two things:
a. T_0^1(f)\leq \lim\inf_{n\rightarrow \infty} T_0^1(f_n), and
b. if each f_n is absolutely continuous and T_0^1(f_n)\leq 1 for each n, then T_0^1(f) = \lim_{n\rightarrow \infty} T_0^1(f_n).
Homework Equations
We denote T_0^1(f) as the total variation of f on [0,1].
The Attempt at a Solution
a.
If f_n is not of bounded variation, then T_0^1(f_n)=\infty and we're done.
So assume f_n is of bounded variation. Then since f_n(x) \rightarrow f(x) for each x, then for a partition 0=t_0 < t_1 < ... < t_N= 1,
|f_n(t_j)-f_n(t_{j-1})| \rightarrow |f(t_j)-f(t_{j-1})|.
So \sum_j|f_n(t_j)-f_n(t_{j-1})| \rightarrow \sum_j |f(t_j)-f(t_{j-1})|.
Take the sup from both sides and so we have T_0^1(f_n)\rightarrow T_0^1(f).
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...