Let $Z$ be a Polish space, and let $\mathcal{P}(Z)$ denote the set of Borel probability measures on $Z$, equipped with the topology of [weak convergence](/page/Weak%20Convergence). For a family $\mathcal{C} \subset \mathcal{P}(Z)$, the following are equivalent:
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1. $\mathcal{C}$ is relatively compact in $\mathcal{P}(Z)$ for weak convergence.
2. $\mathcal{C}$ is tight, meaning that for every $\varepsilon > 0$ there exists a compact set $K_\varepsilon \subset Z$ such that