Definition 3

Let T be a locally compact space, $\mu$ a positive measure on T. An equivalence relation R in T is said to be $\mu$-measurable if there exists a $\mu$-dense set of compact subsets K of T for which the relation $R_K$ is Hausdorff.

If R is Hausdorff then R is $\mu$-measurable, because if $\varphi$ is the canonical mapping of T onto the Hausdorff topological space T/R, $\varphi$ is continuous and R is equivalent to $\varphi(x) = \varphi(y)$. Similarly, if R is such that the saturation for R of every compact subset of T is closed (in particular, if R is a closed equivalence relation), then R is $\mu$-measurable, because for every compact subset K of T, the relation $R_K$ is closed, hence Hausdorff (GT, I, §10, No. 4, Prop. 8).

Note that if R is $\mu$-measurable, then R is also measurable for every measure on T with base $\mu$.