@mike Thank you for considering my question. I am not sure I understand your comment. For example, suppose $f_n$ is defined on $(0,1)$ as follows: $f_n(x)=n-1$ if $x\in (1-1/(2n),1+1/(2n))$ and equals $1/(n-1)$ otherwise. I think $f_n$ is a density on $(0,1)$ with full support and converges weakly to the delta function at $1/2$. But I may well be confused.