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

## TL;DR

This paper introduces two measure-theoretic notions of local conditional entropy for finite covers conditioned on partitions, proves their equivalence, and extends a variational principle to open covers, advancing the theoretical understanding of entropy in measure theory.

## Contribution

It defines and proves the equivalence of two measure-theoretic conditional entropy notions and extends a variational principle to open covers.

## Key findings

- The two notions of conditional entropy are shown to be equal.
- A local variational principle for open covers is established.
- The work extends previous results in measure-theoretic entropy theory.

## 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}.