A Tight Family of Finite Borel Measures Has an Increasing Sequence of Compact Sets Carrying All but 1/j of Every Mass

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 tight family of finite Borel measures on $S$.

(i) Increasing compact sets carrying almost all of every mass. There is a sequence $(K_{j})_{j\in\mathbb{N}}$ of subsets of $S$, each compact in $(S,\mathcal{T}_{d})$, such that $K_{j}\subseteq K_{j+1}$ for every $j\in\mathbb{N}$ and

$$\mu_{i}(S\setminus K_{j})<\frac{1}{j}\qquad\text{for all }i\in I\text{ and }j\in\mathbb{N}.$$