Revista Matemática Iberoamericana


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Volume 23, Issue 3, 2007, pp. 811–830
DOI: 10.4171/RMI/514

Published online: 2007-12-31

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

Esa Järvenpää[1], Maarit Järvenpää[2], Kevin Rogovin[3], Sari Rogovin[4] and Nageswari Shanmugalingam[5]

(1) University of Jyväskylä, Finland
(2) University of Jyväskylä, Finland
(3) University of Jyväskylä, Finland
(4) University of Jyväskylä, Finland
(5) University of Cincinnati, USA

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.

Keywords: $\mathrm{MEC}_p$-space, analytic set, doubling measure, weak $p$-Poincaré inequality, quasi-convexity

Järvenpää Esa, Järvenpää Maarit, Rogovin Kevin, Rogovin Sari, Shanmugalingam Nageswari: Measurability of equivalence classes and MEC$_p$-property in metric spaces . Rev. Mat. Iberoam. 23 (2007), 811-830. doi: 10.4171/RMI/514