Let $(\Omega,\mathcal{A}, P)$ be a probability space and $X$ be a Borel measurable and separable map.

(i) $X_{n}\stackrel{\text{ as }}{\rightarrow}X$ and $\left(d\left(X_{n},X\right) \right)$ is asymptotically measurable;

(ii) $X_{n}\stackrel{\text{ as }}{\rightarrow}X$ and $\left( X_{n} \right)$ is asymptotically measurable.

How can we show that (i) implies (ii)?

(This is a part of Problem 1.9.1 on van der Vaart and Wellner, 1996)

Here, $X_{n}\stackrel{\text{ as }}{\rightarrow}X$ means that $P_{*}\left(\lim d\left(X_{\alpha}, X\right)=0\right)=1$.

$\left( X_{n} \right)$ is asymptotically measurable if and only if $\mathrm{E}^{*} f\left(X_{\alpha}\right)-\mathrm{E}_{*} f\left(X_{\alpha}\right) \rightarrow 0, \quad \text { for every } f \in C_{b}(\mathbb{D})$.