Scheffé’s theorem

Let $X,X_1,X_2,\mathrm{\dots }$ be continuous random variables in a probability space, whose probability density functions are $f,f_1,f_2,\mathrm{\dots }$, respectively. If $f_n\to f$ almost everywhere (relative to Lebesgue measure,) then $X_n$ converges to $X$ in distribution (http://planetmath.org/ConvergenceInDistribution):$X_n\stackrel{𝐷}{\to }X$.