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In this paper, we introduce two measure theoretical notions of conditionalentropy for finite measurable covers conditioned to a finite measurablepartition and prove that they are equal. Using this we state a localvariational principle with respect to the notion of conditional entropy definedby Misiurewicz in \cite{M} for the case of open covers. This in particularextends the work done in \cite{R} and \cite{HYZ}.

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