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?


You just asked this in response to my answer to this question, but you didn't give me a chance to answer it there!

By expressing the problem in this very concrete way, you've made it harder. It is much easier to understand if you look at it abstractly. Let $X_r$ be the subspace of $L^2(\mathbb{R})$ consisting of the functions which are constant on each of these intervals $\alpha + J_r$, and let $P_r$ be the orthogonal projection from $L^2(\mathbb{R})$ onto $X_r$. Then $f_n = P_{r_n}(\chi_E)$, so you are asking whether $P_{r_n} \chi_E \to \chi_E$.

But as you notice (and as I said in my previous answer), this is easy when $E$ is an interval. Taking linear combinations, we get $P_{r_n}f \to f$ whenever $f$ is a finite linear combination of characteristic functions of intervals. But these functions are dense in $L^2(\mathbb{R})$, so we finally get $P_{r_n}f \to f$ for any $f \in L^2(\mathbb{R})$ by an $\frac{\epsilon}{3}$ argument.

  • $\begingroup$ (In fact you can answer there but I'm afraid the answer may be long and not suitble in the comments so I start a new question.) I think finite linear combinations of characteristic functions of intervals are not dense in $L^2 ( \mathbb{R})$, what I know is that finite linear combinations of characteristic functions of Lebesgue measurable sets, i.e. simple functions, are dense in $L^2 (\mathbb{R})$. $\endgroup$ – Lao-tzu Feb 21 '16 at 3:37
  • $\begingroup$ Continuous functions with compact support are dense in $L^2(\mathbb{R})$, and finite linear combinations of characteristic functions of intervals can approximate any continuous function with compact support. $\endgroup$ – Nik Weaver Feb 21 '16 at 3:58
  • $\begingroup$ Nice argument, thank you very much! $\endgroup$ – Lao-tzu Feb 21 '16 at 4:00
  • $\begingroup$ Or from basic measure theory, if $E$ is any Borel subset of $\mathbb{R}$ then we can find a finite union of intervals whose symmetric difference with $E$ has arbitrarily small measure. That gets you to $\chi_E$ directly. $\endgroup$ – Nik Weaver Feb 21 '16 at 4:19
  • $\begingroup$ You're welcome! $\endgroup$ – Nik Weaver Feb 21 '16 at 6:32

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