Let $(X\_{n})\_{n \in \mathbb{N}}$ be a sequence of uniformly integrable random vectors with values in some normed vector space $V$ with $\mathbb{E}[\|X\_n\|] < \infty$. This means that
$$
\lim\_{C \to \infty} \sup\_{n \in \mathbb{N}} \mathbb{E}[ \| X\_{n} \| \mathbb{1}\_{\{\| X\_{n} \| > C\}} ] = 0.
$$
Furthermore, suppose that $\mu$ is a probability measure on $V$ and the sequence $(X\_{n})\_{n \in \mathbb{N}}$ converges weakly to $\mu$. This means that for every bounded Lipschitz continuous function $f \colon V \rightarrow \mathbb{R}$ it holds 
$$
\lim\_{n \rightarrow \infty} \int\_{V}f dP\_{X\_{n}} = \int\_{V} f d\mu.
$$
How can I show that then $\int\_{V} \| x \| d\mu(x) < \infty$?