in my script that I use for college we have the Lebesgue Theorem two times - one for segment sets and the other for rectangle sets.
Just in case, the Lebesgue Theorem says that a function from $[a, b]$ to $\mathbb{R}$ (from $[a, b]\times[c, d]$ to $\mathbb{R}$ in the case with a rectangle) is integrable if and only if its set of discontinuities' Lebesgue measure is zero.
I'm interested to know if there is a generalisation of this theorem. I.e. does it work if you just put the domain as an n-dimensional rectangle?