Theorem 1 (Lusin’s theorem). Suppose that $X$ and $Y$ are Polish spaces, that $\mu$ is a finite Borel measure on $X$, that $f:X\to Y$ is Borel measurable, and that $\varepsilon>0$. Then there exists a compact subset $K$ of $X$, with
$$
\mu(X\setminus K)<\varepsilon,
$$
such that the restriction of $f$ to $K$ is continuous.

Definition 1. Suppose that $f$ is a function on the Borel subsets of a metric space $X$ taking values in $[0,\infty]$. We say $f$ is locally finite if for each $x \in X$ there exists a neighborhood $N$ of $x$ with $f(N) < \infty$.

Definition 1. A Lie algebra $L$ is abelian if $[L,L]=0$, which means for all $x,y\in L,\ [x,y]=0$.

Definition 1. A mapping $f$ from the Borel sets of a metrizable space $(X,\tau)$ to $[0,\infty]$ is tight if $f(K)<\infty$ for each compact $K$ in $X$ and
$$
f(A)=\sup\{f(K):K \text{ compact},\ K\subseteq A\},
\quad \text{for each } A\in \mathcal{B}(X).
$$

Throughout this article $(X,d)$ will be a given complete metric space.

Representations

Definition 1. Let $M_n(k)$ be the algebra of $n\times n$ matrices over $K$ and let
$$
\mathfrak{gl}_n(K):=[M_n(K)]
$$

be the corresponding Lie algebra, which is also called the general linear Lie algebra.

Lie Algebra

Definition 1. A Lie algebra is a vector space $L$ over a field $K$ with a Lie bracket
$$
\begin{aligned}
\left[ \cdot,\cdot \right] &: L \times L \to L\\\
(x,y) &\mapsto [x,y],
\end{aligned}
$$
such that

Definition 1. If $(X,\tau)$ is a topological space, then the Borel $\sigma$-field $\mathcal{B}$ of $X$ is the $\sigma$-field generated by the open sets of $X$ and we say a measure $\mu$ defined on $(X,\mathcal{B})$ is a Borel measure.

Let $\mathcal{P}_2(\mathbb{R}^d)$ denote the space of probability measures on $\mathbb{R}^d$ with finite second moment and $F: \mathcal{P}_2(\mathbb{R}^d)\to \mathbb{R}$ be the potential functional. We are interested in the following optimization problem

$$
\min_{m\in \mathcal{P}_2(\mathbb{R}^d)} F(m).
$$

Motivation

Let $(X,d)$ be a metric space, $F, F_n: X\to \overline{\mathbb{R}}, n=1,2,\cdots$ be functionals, suppose that $x_n\in X$ minimizes $F_n$ for each $n=1,2,\cdots$, does $\lim\limits_{n\to\infty} x_n$ (if it exists) minimize any functional $F$? And in what sense does $F_n$ converge to $F$ ensure the minimizer of $F_n$ converges to minimizer of $F$?