 Solution

The original PDF has height $1/n$ on the interval $[0,e^n]$. For finite $p$,so the sequence cannot converge in $L^p$, since a norm-convergent sequence is bounded. For $p=\infty$, $\Vert f_n{\Vert }_{\infty }=1/n\to 0$, so it converges to zero in $L^{\infty }$. It also converges pointwise, and hence almost everywhere, to zero. Finally for each $ϵ>0$, once $n>1/ϵ$ the set $\{|f_n|>ϵ\}$ is empty. Thus it converges in measure to zero, even on this infinite-measure space.