Limit of translates of characteristic function

Question

This might be silly, but what is a simple way of showing that given a characteristic function of a lebesgue measurable set in $\mathbb{R}$ then we have $\lim_{t \rightarrow 0} \chi (x-t) \rightarrow \chi(x)$ almost everywhere? This looks obvious, but is there an easy way to prove it?

Answer 1

Observe that $\{\lim_{t \rightarrow 0} \chi_A (x-t) \nrightarrow \chi_A(x)\}$ is exactly the boundary of $A$, so the convergence holds a.e. iff the boundary has zero measure. Which is not always the case - example.