A note on maximal conditional entropy on Lebesgue spaces

TL;DR

This paper investigates the conditions under which maximal conditional entropy becomes infinite in Lebesgue spaces, extending Martin's 1969 results on martingale convergence by analyzing information conveyed by sub $\sigma$-fields.

Contribution

It provides new criteria for when the maximal conditional entropy is infinite in Lebesgue spaces, complementing existing martingale convergence results.

Findings

Identifies conditions leading to infinite maximal conditional entropy in Lebesgue spaces.

Shows that excessive information in the sub $\sigma$-field causes divergence of entropy.

Includes an example involving continuous functions with compact support.

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.