Why do we use the Borel sigma algebra for the codomain of a measurable function? In several measure theory books, I see that a measurable function $f:\mathbb{R} \rightarrow \mathbb{R}$ often equips the domain with the Lebesgue $\sigma$-algebra and the codomain with the Borel  $\sigma$-algebra. However, if $g$ is another such function, then the composition of $f$ and $g$ may fail to be measurable. 
Why do we not use the Lebesgue $\sigma$-algebra for both the domain and the codomain?

Good question; I didn't know the answer until just now, never having thought about it.