A note on maximal conditional entropy on Lebesgue spaces

Abstract

Let $(X,\mathcal{B},P)$ be a probability space and $a$ be a sub $\ddot{I}ƒ$-field that is generated by an increasing sequence of sub $\ddot{I}ƒ$-fields $(a_n{)}_{n\in \mathbb{N}}$. Given $\hat{I}¸\in \hat{I}˜$, where $\hat{I}˜$ is some set, let $(X_n^{\hat{I}}¸{)}_{n\in \mathbb{N}}$ be a martingale adapted to $(a_n{)}_{n\in \mathbb{N}}$. Martin (1969) provides sufficient conditions to show that $(X_n^{\hat{I}}¸{)}_{n\in \mathbb{N}}$ converges a.s. uniformly on $\hat{I}˜$ to a random variable $X^{\hat{I}}¸$. His results are based on the assumption that there exists an integer $n$ s.t. the conditional entropy given $a_n$ is uniformly bounded over the set of finite partitions of $X$ with atoms from $a$. This study complements Martin's results by studying the latter assumption on the maximal conditional entropy in the context of measurable partitions of Lebesgue spaces. We provide conditions under which $a$ conveys too much information for the maximal conditional entropy to be finite. As an example, we consider the space of continuous functions with a compact support, equipped with the Borel $\ddot{I}ƒ$-field.