Definition (Relatively Sequentially Compact)

Let $X$ be a Metric Space and let $\mathcal{B}$ be the Borel σ-algebra of $X$. Let $\Pi$ be a family of Probability Measures on $(X,\mathcal{B})$ endowed with the weak topology. We call $\Pi$ relatively sequentially compact if every sequence of elements of $\Pi$ contains a weakly convergent subsequence; that is, if $\forall\{P_{n}\}\subset\Pi$, $\exists\{P_{n_{i}}\}$ and a probability measure $Q$ (defined on $(X,\mathcal{B})$ but not necessarily an element of $\Pi$) such that $$P_{n_{i}}\to Q\text{ weakly}$$