Can we approach the Riemann integral with measure theory?
That is: can we find a measure $\mu$ defined on a $\sigma$-algebra of $\mathbb{R}$ such that a function is $\mu$-integrable if and only if it is Riemann integrable, and that the integral $\int f d\mu$ is equal to the corresponding Riemann integral. If so can we extend this to improper Riemann integrals? What about Riemann integration in $\mathbb{R}^n$?