The measure half of the $2n+1$ problem

Let $F_0,\dots,F_{n-1}$ be Borel functions on a standard Borel space $X$ and let $G_F$ be the graph they generate. We prove that for every Borel probability measure $μ$ on $X$ there are a forward-invariant $μ$-conull Borel set $A \subseteq X$ and a proper Borel $(2n+1)$-coloring of the restriction o