Borel measurable function defined on level sets

Question

Let $(X,\mu)$ be a $\sigma$-finite measure on a standard Borel space $X$ (meaning that $X$ is a separable, complete metric space). Assume that $F:X \rightarrow \mathbb{R}$ and $h:X \rightarrow \mathbb{R}$ are Borel measurable functions with the following property: there exists a conull subset $Y \subset X$ such that $h(x) = h(y)$ implies $F(x) = F(y)$ for all $x,y \in Y$.

Does there exist a Borel measurable function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $F = f \circ h$ almost everywhere on $X$?
One possible strategy is to define the function $f$ on $h(Y)$ as follows: $f(h(x)) = F(x)$, which is well-defined. However how to extend $f$ to a measurable function on $\mathbb{R}$? Note we do not know that $h(X)$ is Borel measurable.