Replacing a sequence of sets by a sequence of disjoint sets

Let \((E_j)_{j\in \NN}\) be a sequence of subsets of a set \(X\), and put \[ F_j\coloneqq E_j \setminus \bigcup_{k<j} E_k . \] The sets \(F_j\) are pairwise disjoint, \(F_j\subseteq E_j\), and \(\bigcup_{j\le n} F_j=\bigcup_{j\le n} E_j\) for every \(n\), so \(\bigcup_j E_j = \coprod_j F_j\). If each \(E_j\) belongs to a \(\sigma\)-algebra \(\mcm\) on \(X\), then so does each \(F_j\).