The functions $n\chi_{(0,1/n)}$ converge to $0$ at every point of $[0,1]$ while each has integral $1$.Suggest a slogan
For \(n\geq1\) let \(f_n\coloneqq n\chi_{(0,1/n)}\colon[0,1]\to\RR\). Then \(f_n(x)\to0\) for every \(x\in[0,1]\), but \[ \lim_{n\to \infty} \int_0^1 f_n(x) \,dx = 1 \neq 0 = \int_0^1 \lim_{n\to \infty} f_n(x) \,dx . \]