I have a sequence of absolutely continuous probability measures $\mu_{n}$ with finite second moment (ie. $\mu_{n}\in P_{ac}(\mathbb{R})\cap P_{2}(\mathbb{R})$), with densities $\rho_{n}\in L^{\infty}(\mathbb{R})$, converging weakly to a probability measure $\mu$ (also with finite second moment) whose density $\rho$ is in $L^{\infty}(\mathbb{R})$. Can I deduce a uniform $L^{\infty}$ bound of the densities, i.e $\|\rho_{n}\|_{L^{\infty}(\mathbb{R})}\leq C$ for some positive constant $C$?
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No. The densities $\rho_n$ could be $1-\varepsilon_n$ on $[0,1]$, $0$ elsewhere expect for a very thin but high peak near $x=n$. Then the measures converge weakly to the uniform distribution on $[0,1]$ but the densities are not uniformly bounded.
answered Nov 30, 2017 at 12:59
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$\begingroup$ Thank you very much Jochen, it seems to be a really nice counterexample! Best, Bruno $\endgroup$ Nov 30, 2017 at 18:07