(X, T) be a topological dynamical system and let µ 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+µ (U, T) and h−µ (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 µ can be defined using either h+µ or h−µ. We also prove that both h+µ and h−µ have an ergodic decomposition and we use it to provea local Abramov formula for h−µ.