Theorem

grouping lemma

Let $\{\mathcal{B}_t, t \in T\}$ be an independent family of $\sigma$-fields. Let $S$ be an index set and suppose for $s \in S$ that $T_s \subset T$ and $\{T_s, s \in S\}$ is pairwise disjoint. Now define $\mathcal{B}_{T_s} = \bigvee_{t \in T_s} \mathcal{B}_{t}$. Then $\{\mathcal{B}_{T_s}, s \in S\}$ is an independent family of $\sigma$-fields. (Remember that $\bigvee_{t \in T_s}\mathcal{B}_t$ is the smallest $\sigma$-field containing all the $\mathcal{B}_{t}$'s.)