why does monotone convergence theorem does not apply on riemann integrals?

I had just learned in measure theory class about the monotone convergence theorem in this version:
for every monotonically increasing series 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 Lebesgue integral, but I didn't find 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)

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