The Kullback-Leibler information function for infinite measures

Abstract

In this paper, we introduce the Kullback-Leibler information function ρ(ν, μ) and prove the local large deviation principle for σ-finite measures μ and finitely additive probability measures ν. In particular, the entropy of a continuous probability distribution ν on the real axis is interpreted as the exponential rate of asymptotics for the Lebesgue measure of the set of those samples that generate empirical measures close to ν in a suitable fine topology.