$\lim_{n \to \infty} \int_{a}^{b}|f_n(x)-f(x)|dx = 0$ implies $f_n \to f$ on $[a,b]$

Question

If $f_n,f$ are Riemann integrable functions on $[a,b]$, if $\lim_{n \to \infty} \int_{a}^{b}|f_n(x)-f(x)|dx = 0$, then is it true that $f_n \to f$ pointwise on $[a,b]$?

Answer 1

No. Let $f_{n}(0)=1$ and $f_{n}(x)=0$ for $x\in(0,1]$ and $f=0$.