Suppose that $\{\mu\_n\}$ is a sequence of Borel probability measures on a compact metric space $X$ and suppose that $\{\mu\_n\}$ converges weakly to a Borel probability measure $\mu$ on $X$. If $\mu$ is absolutely continuous with respect to $\mu^\*$ (where $\mu^\*$ is a Borel probability measure on $X$ different from $\mu$), does there exist a sequence $\{\mu^\*\_n\}$ of Borel probability measures on $X$ such that for each $n$, $\mu\_n$ is absolutely continuous with respect to $\mu^\*\_n$ and the sequence $\{\mu^\*\_n\}$ converges weakly to $\mu^\*$?



Convergence of measures to an absolutely continuous measure
Yes. Take $\mu^\*\_n=(1-1/n)\mu^\* + (1/n)\mu\_n$. 


