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

Abstract: 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 in \cite{M} for the case of open covers. This in particular extends the work done in \cite{R} and \cite{HYZ}.