If $f : [c,d] \to \Bbb{R}$ and $g : [a,b] \to [c,d]$ are functions of bounded variation, then $f \circ g$ is measurable.
I am having trouble with proving the above claim. Here are my thoughts. Recall that a function $f$ is of bounded variation if and only if it is the difference of two increasing functions. Since increasing functions are measurable, so are differences of them. Furthermore, increasing functions are continuous a.e., so differences of them are also continuous a.e. This means that $g$ is measurable and that $f$ is continuous a.e. From this I would like to conclude that $f \circ g$ is measurable, but I don't know if that actually follows.
A monotonic function is Borel measurable, hence $f,g$ are Borel measurable. And the composition Borel measurable functions is Borel measurable.