Exploiting Christian Remling's idea, we can take the following example: $$f_n:=b_n\mathbb 1\left(\bigcup_{j=1}^n\left(j /n-a_n,j/n+a_n\right) \right),$$ where the sequence $(a_n)$ is such that the series $\sum_n na_n $ converges and $na_nb_n=1$ for each $n$. Since for a continuous function $h$, the inequality $$\left|\int f_nh\mathrm dx-\frac 1n\sum_{j=1}^nh(j/n)\right| \leqslant nb_na_n \sup_{\substack{ s,t\in[0,1]\\ |s-t|\leqslant a_n } }|h(t)-h(s)|= \sup_{\substack{ s,t\in[0,1]\\ |s-t|\leqslant a_n } }|h(t)-h(s)|, $$ hence we have $$\lim_{n\to \infty}\int f_nh\mathrm dx=\int h\mathrm dx.$$

Since $\sum_n\lambda\{x\mid f_n(x)\neq 0\}\leqslant \sum_nna_n$, we have $f_n\to 0$ almost everywhere (by the Borel-Cantelli lemma).

Exploiting Christian Remling's idea, we can take the following example: $$f_n:=b_n\mathbb 1\left(\bigcup_{j=1}^n\left(j /n-a_n,j/n+a_n\right) \right),$$ where the sequence $(a_n)$ is such that the series $\sum_n na_n $ converges and $na_nb_n=1$ for each $n$. Since for a continuous function $h$, the inequality $$\left|\int f_nh\mathrm dx-\frac 1n\sum_{j=1}^nh(j/n)\right| \leqslant nb_na_n \sup_{\substack{ s,t\in[0,1]\\ |s-t|\leqslant a_n } }|h(t)-h(s)|= \sup_{\substack{ s,t\in[0,1]\\ |s-t|\leqslant a_n } }|h(t)-h(s)|, $$ hence we have $$\lim_{n\to \infty}\int f_nh\mathrm dx=\int h\mathrm dx.$$

Since $\sum_n\lambda\{x\mid f_n(x)\neq 0\}\leqslant \sum_nna_n$, we have $f_n\to 0$ almost everywhere (by the Borel-Cantelli lemma).