Let $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z} \subset\mathbb{R} (r>0)$, let $E\subset\mathbb{R}$ be a Lebesgue measurable set with finite measure $|E|$, define $J_r=(-\frac{1}{4\pi r}, \frac{1}{4\pi r}), J_r(\alpha)=\alpha+J_r$ for $\alpha\in\Lambda_r^*$, $\{r_n\}$ any positive sequence with $r_n\to\infty$ as $n\to \infty$. Define $$f_n =\sum_{\alpha\in\Lambda_{r_n}^*} 2\pi r_n |E\cap J_{r_n}(\alpha)|\chi_{J_{r_n}(\alpha)}.$$

Then is it true that $f_n \to \chi_E$ (a.e. pointwise or in $L^2$ sense) as $n\to \infty$?

I think it's easy when $E$ is an interval, but I can't prove it for general $E$. Can any one give some help?