a separating class

Before asking whether a small family of test functions can detect CONVERGENCE, ask the more basic question: can it tell two different measures apart at all? A class of functions that can is called separating (or measure-determining). It is the weaker, static cousin of a convergence-determining class: separating is about distinguishing fixed measures, while convergence-determining is about forcing a limiting process.

A class M of bounded measurable (usually continuous) functions on S is separating, or measure-determining, if integral f dmu = integral f dnu for all f in M implies mu = nu, for probability measures mu, nu. Equivalently, M separates the points of P(S): no two distinct probability measures give the same integrals against everything in M. Classic separating classes: on R the indicators 1_{(-infinity, x]} (a CDF determines a law); the moments x^k, but ONLY when the moment problem is determinate (e.g. Carleman's condition holds), since the lognormal has the same moments as a whole family of distinct distributions; the exponentials e^{itx} (characteristic functions are always separating, by Fourier inversion). On any metric space, C_b(S) itself is separating, that is the statement that integrals against all bounded continuous functions pin down a Borel measure.

Separating is exactly what you need to IDENTIFY a weak limit once you already know one exists. The standard pattern: tightness (via Prokhorov) gives a convergent subsequence; you compute integral f dmu_n for f in a separating class and find it converges to integral f dmu_*; separation then forces every subsequential limit to equal mu_*, so the whole sequence converges to mu_*. Thus separating + tight = convergence-determining-in-effect. The honest warning is the moment problem: moments form a separating class only when the limiting distribution is determined by its moments; for heavy tails this can fail and 'matching all moments' does not pin down the law.