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Is there some kind of measure theory generalized to uncountable unions? Of course one needs to take care how to make sense of sums over an uncountable index set. I was thinking about following formulation of the additivity property of the "measure":

$$\mu\left(\bigcup_{i\in\ I} A_i\right)=\sup_{J\subset\ I}\sum_{i\in\ J}\mu\left(A_i\right)$$

Here ##I## can be uncountable, but the supremum is taken over all countable subsets ##J##.

Obviously, if we allow all uncountable unions on ##P(\mathbb{R})##, only the trivial measure can satisfy this (because we can divide every set into subsets of single elements with "measure" 0). But can there be a (nontrivial) subset of uncountable unions that allows for a nontrivial "measure"?