Let $(X,\mathcal B,\mu)$ be a probability space, and let $\alpha$ and $\beta$ be finite measurable partitions of $X$. Assume that $\alpha$ refines $\beta$ modulo $\mu$-null sets, written $\alpha \succeq \beta$: for every atom $B \in \beta$, there is a subfamily $\alpha_B \subset \alpha$ such that