Local conditional entropy in measure for covers with respect to a fixed partition

In this paper we introduce two measure theoretical notions of conditional entropy for finite measurable covers conditioned to a finite measurable partition and prove that they are equal. Using this we state a local variational principle with respect to the notion of conditional entropy defined by Misiurewicz (1976 Stud. Math. 55 176-200) for the case of open covers. This in particular extends the work done in Romagnoli (2003 Ergod. Theor. Dynam. Syst. 23 1601-10), Glasner and Weiss (2006 Handbook of Dynamical Systems vol 1B (Amsterdam: Elsevier)) and Huang et al (2006 Ergod. Theor. Dynam. Syst. 26 219-45).