Let $B$ be the set of bounded functions $[0, 1] \rightarrow \mathbb R$. a sequence $f_n$ of $B$ converges weakly to a distribution $\mu$ if $$\int_0^1 g(t) f_n(t) \rm{d}t \rightarrow \int_0^1 g d\mu \quad \text{ for all continuous functions } g.$$
Under what conditions is $\mu$ absolutely continuous w.r.t the Lebesgues measure? (i.e. $\exists f \in B \text{ s.t } d\mu = f dt$?)
One such a condition is that the $f_n$ be uniformly bounded. Are there others?