  This paper concerns the convergence of empirical measures in high dimensions.
We propose a new class of probability metrics and show that under such metrics,
the convergence is free of the curse of dimensionality (CoD). Such a feature is
critical for high-dimensional analysis and stands in contrast to classical
metrics ({\it e.g.}, the Wasserstein metric). The proposed metrics fall into
the category of integral probability metrics, for which we specify criteria of
test function spaces to guarantee the property of being free of CoD. Examples
of the selected test function spaces include the reproducing kernel Hilbert
spaces, Barron space, and flow-induced function spaces. Three applications of
the proposed metrics are presented: 1. The convergence of empirical measure in
the case of random variables; 2. The convergence of $n$-particle system to the
solution to McKean-Vlasov stochastic differential equation; 3. The construction
of an $\varepsilon$-Nash equilibrium for a homogeneous $n$-player game by its
mean-field limit. As a byproduct, we prove that, given a distribution close to
the target distribution measured by our metric and a certain representation of
the target distribution, we can generate a distribution close to the target one
in terms of the Wasserstein metric and relative entropy. Overall, we show that
the proposed class of metrics is a powerful tool to analyze the convergence of
empirical measures in high dimensions without CoD.
