Let $(\Omega,\mathcal{F},P)$ be a probability space, $\{\mathcal{F}_n \subseteq \mathcal{F}\}_{n \in \mathbb{N}}$ a filtration and $X: \Omega \rightarrow \mathbb{N}$ a random variable. Denote $\mathcal{P}(\Omega)$ the space of probability measures on $(\Omega, \mathcal{F})$ and assume that $\{Q_n: \Omega \rightarrow \mathcal{P}(\Omega)\}_{n \in \mathbb{N}}$ are regular conditional probabilities corresponding to $\mathcal{F}_n$. That is, for any $A \in \mathcal{F}$, $Q_n(\omega)(A)$ is an $\mathcal{F}_n$-measurable function of $\omega \in \Omega$ and for any $B \in \mathcal{F}_n$

$$\int_B Q_n(\omega)(A) P(d\omega) = P(A \cap B)$$

For each $n \in \mathbb{N}$, define the information gain $I_n: \Omega \rightarrow \mathbb{R} \sqcup \{+\infty\}$ by

$$I_n(\omega):= D_{KL}(X_* Q_{n+1}(\omega) \mid\mid X_* Q_{n}(\omega))$$

Here, $X_*$ denotes push-forward by $X$, i.e. $X_* Q_{n}(\omega)$ is the probability measure on $\mathbb{N}$ s.t. for any $A \subseteq \mathbb{N}$

$$X_* Q_{n}(\omega)(A) := Q_n(\omega)(X^{-1}(A))$$

$D_{KL}$ is the Kullback–Leibler divergence, i.e. given probability measures $\mu,\nu$ on $\mathbb{N}$, regarded as functions from $\mathbb{N}$ to $[0,1]$:

$$D_{KL}(\mu \mid\mid \nu) := \sum_{k = 0}^\infty \mu(k) \ln{\frac{\mu(k)}{\nu(k)}}$$

Is it true that $\sum_{n=0}^\infty I_n < \infty$ almost surely?

If it helps, consider any or all of the following simplifying assumptions:

• $\Omega$ is standard.
• $\mathcal{F}_n$ corresponds to a finite partition of $\Omega$.
• $X$ has finite range.