Measure theory is somewhat focused on the cardinal $\aleph_0$: First of all we have the usual $\sigma$-additivity, Polish (separable!) spaces such as $\mathbb{R}^n$, countable sequences and limits, countable recursive constructions. I know that this is a vague question, but is there a possibility to extend measure theory to an arbitrary regular cardinal $\kappa$? Thus in the definitons we replace $\aleph_0$ by $\kappa$ and try to imitate the theory? Perhaps we should also replace $\mathbb{R}$ by some other model because in $\mathbb{R}$ every convergent sum is supported on a countable index subset. Or is there already some research on it?

The reason for my question is not only just of curiosity. I want to understand in detail this "cardinality boundary" of measure theory.