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

We prove that a locally compact metric space that supports a doubling measure and a weak p -Poincaré inequality for some 1\le p 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.