Example for the benefit from monotone convergence

Question

I want to see a (preferably simple) example where I can apply monotone convergence to a sequence of functions $f_n$ but where I cant exchange limitation and integration in terms of the Riemann integral.

Of course this sequence of functions should not be uniformly convergent but the convergence should be monotone.

Answer 1

Let $(q_n)$ be an enumeration of the rationals and consider the sequence of functions defined $f_n(x) = \chi_{q_1, ..., q_n}(x)$. Then $f_n(x) \to \chi_{\mathbb{Q}}(x)$ monotonically so $\lim_n \int_E f_n = \int_E \lim_n f_n = \int_E \chi_{\mathbb{Q}}$ for any measurable set $E$. Note that we cannot draw the same conclusion in the Riemann integrable sense because the functions involved are not even Riemann integrable.