Entropy and information can be considered dual: entropy is a measure of the subspace defined by the information constraining the given ambient space. Negative entropies, arising in na\"ive extensions of the definition of entropy from discrete to continuous settings, are byproducts of the use of probabilities, which only work in the discrete case by a fortunate coincidence. We introduce the notions of sup-normalization and information measures, which allow for the appropriate generalization of the definition of entropy that keeps with the interpretation of entropy as a subspace volume. Applying this in the context of topological groups and Haar measures, we elucidate the relationship between entropy, symmetry, and uniformity.

翻译：熵与信息可视为对偶概念：熵是约束给定环境空间的信息所定义的子空间的度量。将熵的定义从离散情形扩展到连续情形时产生的负熵，是概率论应用中的副产品——概率仅在离散情形下因偶然的巧合而有效。我们引入超归一化与信息度量的概念，这些概念允许对熵的定义进行恰当推广，从而保持熵作为子空间体积的解释。将该方法应用于拓扑群与哈尔测度的语境中，我们阐明了熵、对称性与均匀性之间的关系。