Helly-Bray theorem

Let $F,F_1,F_2,\mathrm{\dots }$ be distribution functions. If $F_n$ converges weakly (http://planetmath.org/ConvergenceInDistribution) to $F$, then

$${\int }_{ℝ}g(x)𝑑F_n(x)\underset{n\to \mathrm{\infty }}{\overset{}{\to }}{\int }_{ℝ}g(x)𝑑F(x)$$

for each continuous bounded function $g:ℝ\to ℝ$.

Remark. The integrals involved are Riemann-Stieltjes integrals.