Abstract

Let (X, T) be a topological dynamical system and let ? be a T-invariant probability measure on X. In this paper, we study two properties of the notions of measure theoretical entropy for a measurable cover U, h?+(U, T) and h?-(U, T) introduced by P. P. Romagnoli (Ergod. Th. & Dynam. Sys. 23 (2003), 1601-1610). The main result of the paper states that entropy pairs for the measure ? can be defined using either h?+ or h?-. We also prove that both h?+ and h?- have an ergodic decomposition and we use it to prove a local Abramov formula for h?-.