TITLE: Definition of Sigma Algebra
QUESTION [6 upvotes]: I was wondering, why are we not allowed to take arbitrary unions (likewise intersections) in the definition of a sigma algebra?; I am looking for a more or less intuitive reason. It seems to me that most of the motivation in defining sigma algebras lies in Measure Theory. So, does allowing arbitrary unions/intersections somehow screw-up the theory?  
Also, is there a mathematical structure which is similar to a sigma algebra, for which arbitrary unions/intersections is specified?
EDIT: By arbitrary, I mean to include families of subsets which are not necessarily index-able by a countable set.

REPLY [3 votes]: If you allowed for unions of families, you can see that this would immediately ruin the theory on $\mathbb{R}$ (and most other interesting places).
This is because any such algebra containing the singletons would also contain any subset of the real numbers. We would hence find ourselves once again at the problem of Carathéodory: having too many measurable sets under which we cannot establish additivity of the form $$\mu(\sqcup E_n) = \sum \mu(E_n).$$