Theorem (Portmanteau): Let $g: \real^d \to \real$. The following conditions are equivalent:

(a) $\x_n \inD \x$.

(b) $\Ex g(\x_n) \to \Ex g(\x)$ for all continuous functions $g$ with compact support.

(c) $\Ex g(\x_n) \to \Ex g(\x)$ for all continuous bounded functions $g$.

(d) $\Ex g(\x_n) \to \Ex g(\x)$ for all bounded measurable functions $g$ such that $g$ is continuous almost everywhere.