# 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 $$K$$.