A characterization of \mu-equicontinuity for topological dynamical systems

Two different notions of measure theoretical equicontinuity ( \mu -equicontinuity) for topological dynamical systems with respect to Borel probability measures appeared in works by Gilman (1987) and Huang, Lee and Ye (2011). We show that if the probability space satisfies Lebesgue's density theorem and Vitali's covering theorem (for example a Cantor set or a subset of \mathbb{R}^{d}), then both notions are equivalent. To show this we characterize Lusin measurable maps using \mu -continuity points. As a corollary we also obtain a new characterization of \mu -mean equicontinuity.