The spikes $n\chi_{(0,1/n)}$ converge pointwise but not in $L^1$

For \(n\geq 1\) let \(f_n\coloneqq n\chi_{(0, 1/n)}\colon\RR\to\RR\).

• For \(x\leq 0\), \(f_n(x) = 0\) for all \(n\); for \(x>0\), \(f_n(x) = 0\) for all \(n > 1/x\). Hence \(f_n\to 0\) pointwise on \(\RR\).

• \(\int_\RR\abs{f_n - 0}\dx = n\cdot\frac1n = 1\) for every \(n\), so \(f_n\not\to 0\) in \(L^1(\RR)\).