If a sequence $(f_n)$ of functions converges to a function $f$ on $S$ (a subset of real numbers), then $f$ is integrable on $S$.

Question

If a sequence $(f_n)$ of functions converges to a function $f$ on $S$ (a subset of real numbers), then $f$ is integrable on $S$.

i know that it is true for uniform convergence is it true for even converges?

Answer 1

Enumerate the rational numbers in $[0,1]$ as $\{r_{1},r_{2},...\}$, let $f_{n}=\displaystyle\sum_{k=1}^{n}\chi_{\{r_{k}\}}$, then $f_{n}\rightarrow\chi_{{\bf{Q}}\cap[0,1]}$ which the latter is not Riemann integrable.