  Two different notions of {\mu}-equicontinuity that apply to topological
dynamical systems and probability measures were studied by Gilman (1987) and
Huang-Lu-Ye (2011). One was used to classify measure preserving topological
dynamical systems and the other for cellular automata. 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 R^{d}) then both properties
are equivalent. To do this we characterize when a function is Lusin measurable.
