I had just learned in measure theory class about the monotone convergence theorem in this version:
For every monotonically increasing sequence of functions $f_n$ from measurable space $X$ to $[0, \infty]$, $$ \text{if}\quad \lim_{n\to \infty}f_n = f, \quad\text{then}\quad \lim_{n\to \infty}\int f_n \, \mathrm{d}\mu = \int f \,\mathrm{d}\mu . $$
I tried to find out why this theorem apply only for a Lebesgue integral, but I didn't find a counter example for Riemann integrals, so I would appreciate your help.
(I guess that $f$ might not be integrable in some cases, but I want a concrete example.)