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.