I want to find a sequence of measurable functions $f\_n(x):[0,1]\rightarrow[0,1]$,such that $\int{f\_n}\rightarrow{0}$ as $n\rightarrow\infty$, but for no $x\in[0,1]$ does $f\_n(x)$ converge to a limit as $n\rightarrow\infty$.

I wanted to use a translation on this sequence and shrink it at the same time, but I got stuck. It always has a lot of points converging to zero as I want to make the integral small. Can anyone help me out?