Let $\sigma$ be an invertible ergodic measure-preserving transformation of a nonatomic standard probability space $(\Omega, \mu)$. For $\omega \in \Omega$ and $j \in \mathbb{N}$ write $I_j(\omega) = \{\sigma^i \omega : -j \le i \le j\}$ for the orbit window of radius $j$.

Let $\varepsilon > 0$, and let $j_1, j_2, \ldots : \Omega \to \mathbb{N}$ be measurable with $j_k(\omega) \to \infty$ for every $\omega$. Then there are a measurable set $A$ and a measurable $j : \Omega \to \mathbb{N}$ such that

• for every $\omega \in A$, $j(\omega) = j_k(\omega)$ for some $k$;
• the windows $I_{j(\omega)}(\omega)$, $\omega \in A$, are pairwise disjoint;
• $\mu\bigl(\bigcup_{\omega \in A} I_{j(\omega)}(\omega)\bigr) > 1 - \varepsilon$.

This is a form of Rudolph's backward Vitali lemma (Rudolph 1990, Theorem 3.9).