Let $(S,d)$ be a Polish space and fix a Borel probability measure $P$. Suppose that $(Q_n)$ is a sequence of Borel probability measures on $S$ such that $Q_n \ll P$ for all $n$ and $Q_n\to Q$ weakly. Is it true that $Q \ll P$?
2026-02-23 06:42:30.1771828950
Weak convergence of probability measures implies absolute continuity with respect a given probability measure?
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Take $(S, d)$ be $[0, 1]$, $Q_n$ to be the uniform distribution on $\left[ -\frac{1}{n}, \frac{1}{n} \right]$ and $P$ to be the Lebesgue measure on $[0,1]$. $Q_n \ll P$ since the uniform distribution has a density function with respect to Lebesgue measure. However which distribution does $Q_n$ converge weakly to?