Show a Carathéodory-constructed measurable set is almost in the underlying algebra
This is easy to do for finite-measure sets, but for the infinite case I can't seem to get a solution...
Let be an algebra of subsets of (i.e. closed under FINITE unions and intersections, and complements, with ). Let be the Carathéodory-constructed measure generated by , and let be a -measurable with . Suppose is -finite by this measure, i.e. that there is a countable sequence with and . I must show that given there is such that . [Recall that .]
As I said, the result holds easily when . But for , I'm stuck. Any help would be much appreciated. Thanks!