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I'm preparing for a test for real analysis and I came across this problem in Royden's book:

Let $\{f_n\}$ be a sequence of real valued functions on $[a,b]$ that converges pointwisely on $[a,b]$ to the real valued function $f$. Show that $TV(f) \leq \liminf ~TV(f_n)?$

This looks quite similar in form to Fatou's Lemma to me, but can't find any way to establish TV with integration, can anybody please help?

(TV is short for total variation)

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2 Answers 2

up vote 2 down vote accepted

Fix $a=t_0<t_1<\ldots<t_N=b$ a subdivision of $[a,b]$. We have $$\sum_{j=1}^N|f(t_j)-f(t_{j-1})|=\lim_{n\to +\infty}\sum_{j=1}^N|f_n(t_j)-f_n(t_{j-1})|,$$ and let $u_n:=\sum_{j=1}^N|f_n(t_j)-f_n(t_{j-1})|$, $v_n:=TV(f_n)$. Since $u_n\leq v_n$ for each $n$, we have $\liminf_{n\to+\infty}u_n=\lim_{n\to+\infty}u_n\leq \liminf_{n\to \infty}v_n$ and we are done.

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Davide and @Nana: Thanks so much, I see how it works now. You guys are awesome! –  Vokram Apr 1 '12 at 4:07


Let $a=x_0\lt \ldots \lt x_n=b$ be a subdivision of $[a,b]$. Let $\varepsilon \gt 0$. Then there is an $M$ such that $$ |f_n(x_k) - f(x_k)|\lt \varepsilon /2,\qquad |f_n(x_{k-1}) - f(x_{k-1})| \lt \varepsilon /2,$$ whenever $M<n$. Then consider

$$\sum_{k=1}^N |f(x_k) - f(x_{k-1})|\leq \sum_{k=1}^N |f(x_k) - f_n(x_{k-1})| +\sum_{k=1}^N |f(x_{k-1}) - f_n(x_{k-1})| \\+\sum_{k=1}^N |f_n(x_k) - f_n(x_{k-1})|.$$

Can you continue?

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