If $f_n: [a,b] \to \mathbb{R}$ are integrable functions and $f_n \to f$ pointwise, then is it true that $f$ is integrable on $[a,b]$ and $$ \lim_{n \to \infty}\int_{a}^{b}f_n(x)\mathrm{d}x = \int_{a}^{b} f(x) \mathrm{d}x $$
I know it holds if $f_n \to f$ uniformly, but not necessarily for pointwise convergence, but cannot think of a counterexample.
Define$$\begin{array}{rccc}f_n\colon&[0,1]&\longrightarrow&\mathbb R\\&x&\mapsto&\begin{cases}0&\text{ if }x<\frac1n\\\frac1x&\text{ otherwise.}\end{cases}\end{array}$$Then $(f_n)_{n\in\mathbb N}$ fonverges pointwise to$$\begin{array}{rccc}f\colon&[0,1]&\longrightarrow&\mathbb R\\&x&\mapsto&\begin{cases}0&\text{ if }x=0\\\frac1x&\text{ otherwise.}\end{cases}\end{array}$$Each $f_n$ is Riemann-integrable, but $f$ isn't. Actually, the same thing occurs with the Lebesgue integral.