 Borel postcomposition preserves Lebesgue measurability

If $g$ is a Lebesgue measurable function and $h$ is a Borel measurable function, then $h\circ g$ is a Lebesgue measurable function. For an open set $U$, $h^{-1}(U)$ is a Borel set. By measurability of $g$, $g^{-1}(h^{-1}(U))$ is a Lebesgue measurable set, and it equals $(h\circ g{)}^{-1}(U)$. This is composition of measurable functions with the intermediate Borel sigma-algebra stated explicitly. Merely knowing both functions are Lebesgue measurable would not ensure that the intermediate sigma-algebras match.