In measure theory, one often wants to show some property $P$ is true of all the sets in some $\sigma$-algebra $\mathcal{B}$. So one sets $\mathcal{A}$ to be all those sets in $\mathcal{B}$ for which $P$ holds, and tries to show $\mathcal{A} = \mathcal{B}$. $\mathcal{A} \subset \mathcal{B}$ is obvious. The reverse is hard to show directly because the sets in $\mathcal{B}$ are usually hard to characterize explicitly. So one often uses something like the Dynkin $\pi$-$\lambda$ theorem. One finds a subset $\mathcal{C} \subset \mathcal{A}$ which is closed under intersection (a $\pi$-system) and with $\sigma(\mathcal{C}) = \mathcal{B}$. Then one shows that $\mathcal{A}$ is a $\lambda$-system (closed under set subtraction and increasing countable union), usually by showing that these operations preserve the property $P$. Using the theorem, one concludes that $\mathcal{B} \subset \mathcal{A}$.

This is really in the spirit of the "closed and dense" technique from topology / analysis.