Tight Family of Borel Measures on a Metric Space

In the setting of Convergence of Probability Measures and Random Elements: Standing Notation, let $I$ be a set and let $(\mu_{i})_{i\in I}$ be a family of finite Borel measures on $S$. For every $K\subseteq S$ that is compact in $(S,\mathcal{T}_{d})$, the set $S\setminus K$ belongs to $\mathcal{B}(S)$ (Closed Subsets of Compact Sets Are Compact, and Compact Subsets of a Metric Space Are Closed and Bounded (ii), Borel Sets of a Metric Space: Open and Closed Sets, Points, Open Sets of Subspaces, and Subspaces (ii)).

(i) Tight family. $(\mu_{i})_{i\in I}$ is tight if for every real number $\varepsilon>0$ there is a set $K\subseteq S$, compact in $(S,\mathcal{T}_{d})$, such that

$$\mu_{i}(S\setminus K)<\varepsilon\qquad\text{for every }i\in I.$$