TITLE: measure theory for regular cardinals
QUESTION [10 upvotes]: 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.

REPLY [2 votes]: A related topic:  A $\sigma$-smooth linear funtional $L$ on $C(X)$, the set of continuous real-valued functions on a topological space $X$, is a linear functional such that: if $f_n$ is a sequence that decreases pointwise to $0$, then $L(f_n) \to 0$.  A generalization is $\tau$-smooth linear functional: if $\mathcal A$ is a family of nonnegative functions in $C(X)$ that is directed in the sense: for any $f_1, f_2 \in \mathcal A$ there exists $f_3 \in \mathcal A$ with $f_3 \le f_1$ and $f_3 \le f_2$, then $\inf_{f \in \mathcal A} L(f) = 0$.  For some topological spaces, every $\sigma$-smooth functional is $\tau$-smooth. For other topological spaces, the two concepts are different.