Ergodicity of Invariant Capacity

TL;DR

This paper explores the concept of ergodicity in the context of capacity preserving transformations, establishing foundational results including a Birkhoff-type theorem and linking ergodicity to the strong law of large numbers.

Contribution

It introduces the notion of ergodicity for capacity preserving transformations and provides characterizations and limit theorems in this framework.

Findings

Existence of a $ heta$-invariant capacity for any measurable transformation

Characterizations of ergodicity in capacity spaces

Ergodicity is equivalent to the strong law of large numbers under upper probability

Abstract

In this paper, we investigate capacity preserving transformations and their ergodicity. We show that for any measurable transformation $\theta$ there always exists a $\theta$-invariant capacity. We investigate some limit properties under capacity spaces and then give the concept of ergodicity for a capacity preserving transformation. Based on this definition, we give several characterizations of ergodicity. In particular, we obtain a type of Birkhoff's ergodic theorem and prove that the ergodicity of $\theta$ with respect to an upper probability is equivalent to the strong law of large numbers.