Category Theory
A category theoretic definition for the equipartition property is given by Gromov Given a sequence of Cartesian powers of a measure space P, this sequence admits an asymptotically equivalent sequence of homogenous measure spaces (i.e. all sets have the same measure; all morphisms are invariant under the group of automorphisms, and thus factor as a morphism to the terminal object) .
The above requires a definition of asymptotic equivalence. This is given in terms of a distance function, giving how much an injective correspondence differs from an isomorphism. An injective correspondence is a partially defined map that is a bijection; that is, it is a bijection between a subset and . Then define
where denotes the measure of a set S. In what follows, the measure of P and Q are taken to be 1, so that the measure spaces are probability spaces. This distance is commonly known as the earth mover's distance or Wasserstein metric.
Similarly, define
with taken to be the counting measure on P. Thus, this definition requires that P be a finite measure space. Finally, let
A sequence of injective correspondences are then asymptotically equivalent when
Given a sequence that is asymptotically equivalent to, the entropy H(P) of P may be taken as
Read more about this topic: Asymptotic Equipartition Property
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