I am reading van der Vaart Wellner's Weak Convergence and Empirical Processes in which they discuss the Portmanteau Theorem (P17-18):
Let $(\Omega_\alpha,A_\alpha,P_\alpha)$ be a net of probability spaces and $X_\alpha:\alpha\to D$ arbitrary maps (D is a metric space). The net $X_\alpha$ converges weakly to a Borel measure L if $$E^*f(X_\alpha)\to \int fdL, \text{for every bounded continuous function on D},$$ where $E^*$ is the outer integral defined as: $$E^*f=inf{EU,f\leq U, U \text{measurable}}$$ They state that the following two conditions are equivalent:
- $X_\alpha$ converges weakly to L
- $\liminf E_*f(X_\alpha)\geq \int f dL$ for every bounded, Lipschitz continuous, nonnegative f.
where $E_*$ is the inner integral defined as $E_*f=-E^*-f$. They say that the proof from 1 to 2 is trivial but I don't know why this is the case. I can prove the theorem using a truncation argument but the proof requires other condition in the Portmanteau theorem. I am just wondering if there is a trivial way to argue the implication.