度量空间的最大熵
The maximum entropy of a metric space
摘要：
我们定义了一个单参数熵族，每个熵族为紧致度量空间（或者更一般地说，具有点之间相似性概念的紧致T2空间）上的任何概率测度分配一个实数。这些熵概括了信息论的香农熵和雷尼熵。我们证明了在任何空间X上，存在一个同时最大化所有这些熵的单一概率测度。而且，所有熵都有相同的最大值：X的最大熵。随着X的放大，最大熵也在增长;它的渐近性决定了X的几何信息，包括体积和维度。我们还研究了极大化测度本身的大尺度极限，认为它应该被看作是X上的规范测度或一致测度。首先，我们不使用熵本身，而是使用它的指数，称为多样性，并（以其有限形式）用作生物多样性的衡量标准。我们的主要定理首先证明了有限的情况下，Leinster和Meckes。
英文摘要：
We define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with a notion of similarity between points). These entropies generalise the Shannon and Rényi entropies of information theory. We prove that on any space X, there is a single probability measure maximising all these entropies simultaneously. Moreover, all the entropies have the same maximum value: the maximum entropy of X. As X is scaled up, the maximum entropy grows; its asymptotics determine geometric information about X, including the volume and dimension. We also study the large-scale limit of the maximising measure itself, arguing that it should be regarded as the canonical or uniform measure on X. Primarily we work not with entropy itself but its exponential, called diversity and (in its finite form) used as a measure of biodiversity. Our main theorem was first proved in the finite case by Leinster and Meckes.
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