TITLE: A Riemann integrable function $f$ on a bounded interval $[a, b]$ is measurable with respect to the Borel measure on $[a,b]$?
QUESTION [6 upvotes]: Suppose $f:[a,b]\rightarrow  [-\infty, \infty]$ is bounded and Riemann integrable, must it be measurable with respect to the Boreal measure on $[a,b]$?

REPLY [9 votes]: The answer is no. We know that a function is Riemann integrable iff it is bounded and a.e. continuous. So if you take $f$ to be the characteristic function of a non-Borel set contained in the standard $1/3$-Cantor set (these sets exist by axiom of choice and a neat construction), then $f$ is Riemann integrable but not Borel measurable. (It is Lebesgue-measurable, though.)