Assume we have a reference measure $\mu$ and an absolutely continuous measure $P$ with Radon-Nikodym derivative $\frac{dP}{d\mu}=p$.
Assume furthermore that we have a sequence of probability measures $Q_n, n=1,2,...$ with Radon-Nikodym derivatives $\frac{dQ_n}{d\mu}=q_n$.
I would like to know if it's possible to establish a "hierarchy" (which one is "stronger") between the following statements:
- $$\mathbb L^1(\Omega)-\lim_{n\to\infty} q_n=p$$
- $$D_{KL}(Q_n\|P)\to 0, \; as\; n\to\infty$$
where $D_{KL}$ denotes the Kullback-Liebler divergence.
Thanks in advance.