pr.probability – Is the set of probability measures on $mathbb{R}$ absolutely continuous with bounded density a closed subset?

The answer is yes. Indeed, a probability measure $mu$ over $mathbb R$ has a density bounded by a real $K>0$ iff the cdf of $mu$ is $K$-Lipschitz, that is, Lipschitz with the Lipschitz constant $K$.

So, you have a sequence $(mu_n)$ of probability measures over $mathbb R$ with $K$-Lipschitz cdf’s $F_n$ converging to the cdf $F$ of a probability measure $mu$ at all points of continuity of $F$. Since the set of all points of continuity of $F$ is dense in $mathbb R$, we conclude that $F$ is $K$-Lipschitz. So, $mu$ has a density bounded by $K$.