MHB Is the integral of a nonnegative Lebesgue integrable function continuous?

[Chris L T521]

Here's this week's problem (and the last Graduate POTW of 2012!).

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Problem: Let $f$ be a nonnegative Lebesgue integrable function. Show that the function defined by\[F(x)=\int_{-\infty}^xf\,dm\]
is continuous by the monotone convergence theorem.

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[Chris L T521]

This week's question was correctly answered by girdav. You can find his answer below:

Spoiler

As $f$ is integrable, we can write it as a limit in $L^1$ of simple functions (that is, linear combinations of characteristic functions, here of sets of finite measure by integrability) $f_n$. This forms a sequence which converges uniformly to $f$ on the real line, so it's enough to do it when $f$ is a simple function. By linearity, it's enough to do it when $f$ is the characteristic function of a measurable set $S$ of finite measure. We have for $s,t\in \Bbb R$ that $|F(s)-F(t)|\leqslant |s-t|\cdot m(S)$, which can be seen assuming for example that $s<t$.