What is the Total Variation of a Convergent Sequence of Functions on [0,1]?

[bham10246]

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...

 

[EnumaElish]

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).

 

[bham10246]

Definition of Total Variation: Let f(t) be complex-valued function defined on the interval [0,1]. Let P: 0=t_0 &lt; t_1 &lt; ... &lt; t_N = 1 be a partition of the unit interval. Then
T_0^1(f) = \sup_{P} \:\sum_{i} |f(t_i)-f(t_i-1)|
where we take the supremum over all partitions of the unit interval.

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