I know that the weak lower semi-continuity of the KL divergence was proved in [1]. If I remember well, the same property is true for any $f$ divergence (with suitable assumptions on the probability space). I am looking for some reference about it.


[1] Posner, Random Coding Strategies for Minimum Entropy, 1975.


**Edit.**  

Here is what I believe a standard definition of $f$ divergences, which includes the case of measures not absolutely continuous to each other. This definition is taken from <http://people.lids.mit.edu/yp/homepage/data/LN_fdiv.pdf>



> 
> **Definition 7.1.** Let $f:(0,\infty)\to\mathbb R$ be a convex function with $f(1)=0$. Let $P$ and $Q$ be two probability distributions on a measurable space $(\mathcal X, \mathcal F)$. If $P\ll Q$ then the $f$-divergence is defined as
> $$D\_f(P\|Q)=\mathbb E\_Q[f(dP/dQ)]$$
> where $dP/dQ$ is the Radon-Nikodym derivative and $f(0)=f(0+)$. More generally, let $f'(\infty)=\lim\_{x\to 0}xf(1/x)$. Let $R$ be such that $Q\ll R$ and $P\ll R$ (such an $R$ always exists, for instance take $R=\frac{1}{2}(P+Q)$. Then we have
> $$D\_f(P\|Q) = f'(\infty)P(dQ/dR=0)+\int\_{dQ/dR>0}\frac{dQ}{dR}f\left(\frac{dP/dR}{dQ/dR}\right)dR\,,$$
> with the agreement that if $P(dQ/dR=0)=0$ the last term is taken to be zero regardless of the value of $f'(\infty)$ (which could be infinite).
> 
> 
> 



Looking for a reference: $f$-divergences are lower semicontinuous
I think the most general reference is:


M. Liero, A. Mielke, and G. Savaré. [Optimal entropy-transport problems and a newhellinger–kantorovich distance between positive measures](https://link.springer.com/article/10.1007/s00222-017-0759-8). *Inventiones mathematicae*, **211**, 2018.  

[arxiv:1508.07941](https://arxiv.org/abs/1508.07941)


Corollary 2.9 proves the lower semicontinuity wrt to the narrow topology under very general assumptions, and Remark 2.1 shows that the weak topology and the narrow topology coincide if $\mathcal X$ is Polish.


