Tight Family of Borel Measures on a Metric Space

Let $(X,d)$ be a metric space, let $\mathcal{T}_{d}$ be the collection of subsets of $X$ that are open in $(X,d)$, which is a topology on $X$ by Metric Open Sets Form a Topology, and let $\mathcal{B}(X)$ be the Borel $\sigma$-algebra of $(X,d)$. Let $\mathbb{R}$ denote the real numbers, with the order $\le$ of their ordered field structure, and for $s,t\in\mathbb{R}$ write $s<t$ to mean that $s\le t$ and $s\ne t$. Let $\mathbb{N}$ be the set of natural numbers.

For every $K\subseteq X$ that is compact in $(X,\mathcal{T}_{d})$, the set $X\setminus K$ belongs to $\mathcal{B}(X)$ by Compact Subsets of a Metric Space are Closed and Borel §borel, so that $\mu(X\setminus K)$ is defined for every Borel measure $\mu$ on $(X,d)$.

1. (Tight family)¶ Let $\mathcal{M}$ be a set whose elements are Borel measures on $(X,d)$. We say that $\mathcal{M}$ is tight in $(X,d)$ if for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is a set $K\subseteq X$, compact in $(X,\mathcal{T}_{d})$, such that

2. (Tight sequence)¶ A sequence $(\mu_{n})_{n\in\mathbb{N}}$ whose terms are Borel measures on $(X,d)$ is called tight in $(X,d)$ if the set $\{\mu_{n}:n\in\mathbb{N}\}$ of its terms is tight in the sense of clause 1.