Example showing the subspace of Riemann integrable functions in $L^2([-\pi,\pi])$ is not closed

Question

In Stein and Shakarski, they say "For instance, as we have already indicated, the subspace of Riemann integrable functions in $L^2([-\pi,\pi])$ is not closed."

Can anybody give me an example of such a sequence?

Answer 1

I am going to transform my comments into an answer.

Let $C_n$ be the closed set that we have at the $n$-th step of the construction of a fat Cantor set $C$ in $[-\pi,\pi]$ and let $f_n=\chi_{C_n}$. $f_n$ is Riemann-integrable and square-integrable and $f_n\to \chi_C$ in $L^2$. However, $\chi_C$ is not Riemann-integrable due to the Lebesgue criterion for Riemann integrability: the set of discontinuities of $\chi_C$ has positive measure.

Answer 2

Also one can do: let $\mathbb{Q}=\{q_{1},q_{2},...\}$. Put $f_{N}(x)=1_{\{q_{1},...,q_{N}\}}(x)$. Each $f_{n}$ is riemman integrable but $f_{n}\to_{\|\cdot\|_{L^{2}}}1_{\mathbb{Q}}$ (it is easy to show that $1_{\mathbb{Q}}$ is not riemman integrable).