Limit of Lebesgue integral over an increment of a function

Question

I am currently reading a proof which uses the following fact which I do not know how to show:

For any function $f \in L^1 (\mathbb{R})$, $$ \lim_{ h \to 0} \int_{\mathbb{R}} |f(x+h) - f(x) | \,dx =0.$$

The main theorems (monotone convergence, dominated convergence and Fatou) do not seem to work. Also, even if we interchange the limit with the integral, we still can't obtain the result, because we don't have continuity in the assumption. Any ideas?

Answer 1

This is easy to show for a compactly supported continuous function, which is necessarily uniformly continuous. Now recall that such functions are dense in $L^1$, so the standard trick of writing

$$|f(x + h) - f(x)| \le |f(x + h) - g(x + h)| + |f(x) - g(x)| + |g(x + h) - g(x)|$$

can be used.