  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}.
