We prove that a locally compact metric space that supports a doubling measure and a weak -Poincaré inequality for some is a -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 -integrable upper gradient is locally -integrable.
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Esa Järvenpää, Maarit Järvenpää, Kevin Rogovin, Sari Rogovin, Nageswari Shanmugalingam, Measurability of equivalence classes and MEC-property in metric spaces . Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 811–830DOI 10.4171/RMI/514