Continuity of a characteristic function of a translated set

Question

Let $E \subseteq \mathbb{R}$ be a measurable set. Is it true that $\chi_{E+t}(x) \rightarrow \chi_{E}(x)$ as $t \rightarrow 0$, where $E+t = \{x+t \, | \, x \in E\}$ for each $t \in \mathbb{R}$, almost everywhere? Is there any counterexample?

Thank you.

Answer 1

Unless I misunderstand -- consider $E = \mathbb{Q}$. Then, since $t$ is irrational infinitely often as it $\to 0$, $\chi_{E+t}$ will be 0 on all of $E$ infinitely often, so it clearly does not converge to $\chi_E$.