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I know that the Kullback-Leibler $D(\mu||\nu) := - \int_K\log\big(\frac{d \nu}{d \mu}\big) \, d\mu,$ over probability measures on a compact $K$ subset of $\mathbb{R}^d$, is only weakly lower semicontinuous.

What I was wondering is whether there are simple characterisations, or at least sufficient conditions for couple of measures to be points of (sequential) continuity.

In the specific problem I have in mind, I'm particularly interested in continuity with respect to $\nu$: I have a sequence $\nu_n$ weakly converging to $\nu$ and I would want to understand cases depending on $\nu$ which allow to infer that the sequence $D(\mu, \nu_n)$ converges to $D(\mu, \nu)$.

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    $\begingroup$ Keep in mind that although the sequence $\nu_ n$ weakly converges to $\nu$, nothing precludes the measures $\nu_n$ from being mutually singular with $\mu$. $\endgroup$
    – R W
    Commented May 31, 2019 at 12:34
  • $\begingroup$ Thanks for the answer. Actually, in my case all the $\nu_n$ are absolutely continuous with respect to $\mu$, and I can even show that all the weak cluster points of the sequence $\nu_n$ also are. $\endgroup$
    – thegain
    Commented Jun 20, 2019 at 18:50

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