The mutual information $I(\mathfrak A\_1;\mathfrak A\_2)$ of two complete $\sigma$-algebras $\mathfrak A\_1$ and $\mathfrak A\_2$ in a Lebesgue probability space $(X,m)$ is the integral of the logarithm of the Radon-Nikodym derivative $dP/d(P\_1\otimes P\_2)$, where $P\_i$ are the quotient measures on the factor-spaces $X\_i$ determined by the $\sigma$-algebras $\mathfrak A\_i$, respectively, and $P$ is the joint distribution on $X\_1\times X\_2$. If $\mathfrak A\_i$ correspond to countable measurable partitions $\alpha\_i$ of the space $X$, then $I(\mathfrak A\_1;\mathfrak A\_2)$ are expressed in terms of the associated entropies as $H(\alpha\_1)+H(\alpha\_2)-H(\alpha\_1\vee\alpha\_2)$. I need an explicit reference for the following fact (I know how to prove it): if $\mathfrak B\_n$ is a decreasing sequence of $\sigma$-algebras converging to a limit $\mathfrak B$, and $\mathfrak A$ is another $\sigma$-algebra such that $I(\mathfrak A;\mathfrak B\_1)<\infty$, then $I(\mathfrak A;\mathfrak B\_n)$ converges to $I(\mathfrak A;\mathfrak B)$.



Continuity of the mutual information
The last proof of this continuity property (and of its analogue for increasing sequences) is given in [this paper](http://www.ams.org/mathscinet-getitem?mr=2472012) by Harremöes and Holst. It also contains a pretty comprehensive list of references to earlier work. Apparently, first this property was established by Pinsker in his 1960 book "Information and Information Stability of Random Variables and Processes".


