Definition 1

Let Y be a topological space. For every measurable function $g$ from X into Y, the $\mu$-essential image of $g$ is the subset of the $y\in Y$ such that, for every open neighbourhood U of $y$, the set $\overset{-1}{g}(U)$ is not locally $\mu$-negligible in X (INT, IV, p. 172, § 5, n$^o2$, def. 3).