A note on maximal conditional entropy on Lebesgue spaces

Abstract

Let $(X,\mathcal{B},P)$ be a probability space and $\mathit{a}$ be a sub $\sigma$-field that is generated by an increasing sequence of sub $\sigma$-fields $(\mathit{a}_{n})_{n \in \mathbb{N}}$. Given $\theta \in \Theta$, where $\Theta$ is some set, let $(X_{n}^{\theta})_{n \in \mathbb{N}}$ be a martingale adapted to $(\mathit{a}_{n})_{n \in \mathbb{N}}$. Martin (1969) provides sufficient conditions to show that $(X_{n}^{\theta})_{n \in \mathbb{N}}$ converges a.s. uniformly on $\Theta$ to a random variable $X^{\theta}$. His results are based on the assumption that there exists an integer $n$ s.t. the conditional entropy given $\mathit{a}_{n}$ is uniformly bounded over the set of finite partitions of $X$ with atoms from $\mathit{a}$. 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 $\mathit{a}$ 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 $\sigma$-field.