Question

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 $I(\mathfrak A;\mathfrak B)$.

Answer 1

The last proof of this continuity property (and of its analogue for increasing sequences) is given in this paper 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".

Answer 2

Mutual information is weak-* lower semicontinuous, because it is Kullback-Leibler divergence. For this see Pinsker's book or Dupuis-Ellis. This gives you the desired liminf. For the other direction (which is usually easy because I haven't used your monotonicity condition yet), maybe you have some monotonicity or convexity argument.