= Borel postcomposition preserves Lebesgue measurability
{c}
{title2=$g:(\mathbb R,\mathcal L)\to(\mathbb R,\mathcal B),\ h:(\mathbb R,\mathcal B)\to(\mathbb R,\mathcal B)$}
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.
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