In this question, I am specifically interested in the following case:
Let f:R->R be a Borel-measurable function from the reals to the reals and let
$g(x)\ =\ \int_{b}^x\ f(t)\ dt$,
where the integral is the Lebesgue integral and b is a real constant. I would presume
$g'(x) = f(x) $ a. e.
Is this true? Is there a published reference for this? (It might be an exercise in a book.) I tried looking in Rao's encyclopedic compendium on Lebesgue measure and integration, among others, but couldn't seem to find an answer.