Maybe we can restrict to one dimension:
When can we integrate a distribution over an interval?

In Lebesgue theory, we can integrate measurable functions. But distributions are not functions. They can be integrated against smooth test functions. So my question is: when can we actually take the test function as indicator function for an interval?

For instance, what is the integral of Dirac at 0 over [0,1] or (0,1)?

Can we integrate something from $H^{-1}$ (Sobolev space) over [0,1]?

Another example of (random) distribution is Gaussian white noise, integrating it over a domain A gives a mean zero Gaussian random variable, whose variance is the area of A. Note that Gaussian white noise is not measurable function...