> [!NOTE] Definition (Common Refinement) > Let $I$ be a [[Closed Real Interval|closed real interval]]. Let $P$ and $P'$ be [[Finite Partition of Closed Real Interval|finite partitions]] of $I.$ We define common refinement $P\#P'$ of $P$ and $P'$ to be the set $P\#P' = \{ K \cap J \mid K\in P \text{ and } J\in P' \}$so that it is a [[Refinement of Finite Partition of Closed Real Interval|refinement]] of both $P$ and $P'.$ ^e69641