Integrating the normal distribution over rational numbers?

Question

Is it possible to integrate the normal distribution over rational numbers? What is the value of such integral? Is it $\pi$ minus the integral over irrational numbers?

Answer 1

The integral of any function over a set with measure $0$ is equal to $0$.

Answer 2

The intersection of $\mathbb{Q}$ with the unit interval $[0,1]$ contains infinitely-many discontinuities (Dirichlet's famous "pathological" function and many similar functions behave strangely/pathologically by taking advantage of $\mathbb{Q}$ as an everywhere dense but incomplete set). It is not Riemann integrable. It is, however, integrable using e.g., Lebesgue integration. But this is done (partly) by realizing that the (set of the degenerate singletons of the) rationals are "negligible" or 0. This can be seen by constructing a function $f(x)$ over the interval $[0,1]$ using the indicator function 1Q (the indicator function for the rationals) or the Dirchlet function $f(x)=1$ if $x$ is rational and $0$ if it isn't (over any interval, including the unit interval). Such functions have an integral of $0$.