As it is known, universal codes, which estimate the entropy rate consistently, exist for any stationary ergodic source over a finite alphabet but not over a countably infinite one. We cast the problem of universal coding into the problem of universal densities with respect to a given reference measure on a countably generated measurable space, examples being the counting measure or the Lebesgue measure. We show that universal densities, which estimate the differential entropy rate consistently, exist if the reference measure is finite, which disproves that the assumption of a finite alphabet is necessary in general. To exhibit a universal density, we combine the prediction by partial matching (PPM) code with the non-parametric differential (NPD) entropy rate estimator, putting a prior both over all Markov orders and all quantization levels. The proof of universality applies Barron's asymptotic equipartition for densities and continuity of $f$-divergences for filtrations. As an application, we demonstrate that any universal density induces a strongly consistent Ces\`aro mean estimator of the conditional density given an infinite past, which solves the problem of universal prediction with the $0-1$ loss for a countable alphabet, by the way. We also show that there exists a strongly consistent entropy rate estimator with respect to the Lebesgue measure in the class of stationary ergodic Gaussian processes.

翻译：众所周知, 一致估算精确率的通用代码, 持续估算精确率的通用代码, 存在于一个固定的字母源中, 使用一个固定的字母, 而不是一个可以计算到的无限值。 我们把通用编码问题放在一个可以计算到的可测量空间上, 在一个可计算到的参考度上, 举例来说, 计数尺度 或 Lebesgue 度量 。 我们显示, 如果参照度量是有限的, 持续估算不同的摄取率, 普遍性的密度存在, 这否定了假设一般需要使用一个限定字母的假设。 要显示一个普遍密度, 我们将部分匹配( PPM) 代码与非参数差异( NPD) 摄取率天平率测量器结合起来, 对所有Markov 订单和所有量度等级都预先设定一个通用的参考度度度度量度。 普遍性的证明, Barron“ 精确” 和 $- distical 调度假设一个非常一致的 Cesqouroal 值 和 最高等级的精确度的测量率 。