Lemma 2

Let $g$ be a continuous function from X into $\mathbf{C}$. The $\mu$-essential image of $g$ is the closure of the image under $g$ of the support of $\mu$.

Let Y be the support of $\mu$. If $z\in \mathbf{C}$ is not adherent to $g$(Y), there exists an open neighbourhood U of $z$ such that the open set $\overset{-1}{g}(U)$ does not meet Y and is therefore locally $\mu$-negligible; this means that $z$ does not belong to the $\mu$-essential image of $g$.

Conversely, if $z\in \mathbf{C}$ is adherent to $g$(Y), then for every open neighbourhood U of $z$, the set $\overset{-1}{g}(U)$ is an open set in X which meets Y. It is therefore not $\mu$-negligible, and since it is open, it is not locally $\mu$-negligible (INT, IV, p. 172, § 5, n$^o2$, Cor. 2). Hence $z$ belongs to the $\mu$-essential image of $g$.