This is easy to do for finite-measure sets, but for the infinite case I can't seem to get a solution...

Let \mathcal{A} be an algebra of subsets of X (i.e. closed under FINITE unions and intersections, and complements, with \emptyset,X\in\mathcal{A}). Let \mu be the Carathéodory-constructed measure generated by \mathcal{A}, and let E be a \mu-measurable with \mu(E)=\infty. Suppose \mathcal{A} is \sigma-finite by this measure, i.e. that there is a countable sequence A_n\in\mathcal{A} with \mu(A_n)<\infty) and \bigcup A_n=X. I must show that given \epsilon>0 there is A\in\mathcal{A} such that \mu(E\Delta A)<\epsilon. [Recall that E\Delta A=(A\cap E^c)\cup(A^c\cap E).]

As I said, the result holds easily when \mu(E)<\infty. But for \mu(E)=\infty, I'm stuck. Any help would be much appreciated. Thanks!