In most courses on Measure Theory the Lebesgue Integral is introduced initially for simple functions on finite spaces, then for general functions on finite spaces and finally for general functions on $\sigma$-finite spaces.
I was wondering if there exists any sort of notion of integrals on non- $\sigma$-finite spaces? If not why so? Is it because there's no way to handle the convergence of non-countable sequences, and we would essentially need to redefine the notion of integrability altogether?
Would appreciate any insight on the matter.
Thanks!