Measurability of equivalence classes and MEC$_p$-property in metric spaces

Abstract

We prove that a locally compact metric space that supports a doubling measure and a weak $p$-Poincaré inequality for some $1\le p < \infty$ is a $\mathrm{MEC}_p$-space. The methods developed for this purpose include measurability considerations and lead to interesting consequences. For example, we verify that each extended real valued function having a $p$-integrable upper gradient is locally $p$-integrable.