The Vitali-Hahn-Saks theorem for algebras (Q1061257)

Let A be an algebra of subsets of a set S. A has the sequential completeness property (SCP) if each disjoint sequence \((A_ j)\) from A has a subsequence \(\{A_{k_ j}\}\) whose union is in A. Theorem. Let A satisfy SCP. If \(\mu_ i:A\to G\) is a sequence of finitely additive, exhaustive set functions such that \(\lim_{i}\mu_ i(A)=\mu (A)\) exists for each \(A_ 0\in A\), then \(\{\mu_ i\}\) is uniformly exhaustive.