Proposition 9

— Let $f$ and $g$ be two continuous mappings of $X$ into a Hausdorff topological space $F$; for $f$ and $g$ to be equivalent, it is necessary and sufficient that $f(x) = g(x)$ at every point of the support of $\mu$.

For, the set of $x \in X$ such that $f(x) \neq g(x)$ is an open set (GT, I, §8, No. 1); for it to be negligible, it is necessary and sufficient that it not intersect the support of $\mu$ (No. 2, Prop. 5).