A note on maximal conditional entropy on Lebesgue spaces

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.