Why countable unions, intersections etc.? I was just wondering why one always insists on countability when it comes to the definition of a $\sigma$-algebra in measure theory. I mean, measure theory works as it does, but is there a deeper reason why e.g. it is reasonable to demand that the countable union of measurable sets is measurable again? Why not an uncountable union? Thanks in advance!

Another reason is that in a summable family $(x_i)_{i\in I}$, it can be shown the set $\;\{i\in I\mid x_i\neq 0\}$  is at most countable. Thus the theory of summable families comes down to the theory of series.
