> [!definition] > > A [[Function|function]] $f: \real \to \mathbb{C}|\real$ is [[Lebesgue Measure|Lebesgue]] measurable if it is $(\lms, \cb_\mathbb{C}|\cb_\real)$ [[Measurable Function|measurable]]. > [!theorem] > > Let $f: \real \to \real$ be a [[Borel Measurable Function|Borel measurable function]] and $g: \real \to \real$ be any function, then $f \circ g$ is Lebesgue measurable if $g$ is Lebesgue measurable. > > *Proof*. Suppose that $g$ is Lebesgue measurable and let $E \in \cb_\real$ be a Borel set, then > $ > \begin{align*} > (f \circ g)^{-1}(E) &= g^{-1}(f^{-1}(E)) \\ > f^{-1}(E) \in \cb_\real &\Rightarrow g^{-1}(f^{-1}(E)) \in \lms > \end{align*} > $ > making $f \circ g$ Lebesgue measurable.