Let $(\mu_n)_{n \in \mathbb N}$ be a sequence of Levy measures such that: $$\lim_{n \to \infty}\mu_n(E) = \mu(E), \quad (\forall\,\, E \,\, \mu-\hbox{Continuity set}, 0 \notin \overline{E})$$ where $\mu$ is another Levy measure. All measures are defined on Borelians of $\mathbb R^m$. Suppose that: $$\int_{\mathbb R^m} |x|^2 \mu_n (dx)\leq C \quad \forall n \in \mathbb N.$$ Given $A=[-a,a]^p = [-a,a] \times \cdots \times [-a,a]$, where $a >0$. I want to show that \begin{equation}\label{I}\tag{I} \int_{A^c}x \mu_n(dx) \longrightarrow \int_{A^c}x \mu(dx), \quad (n \to \infty) \end{equation} Recall that by definition of Levy measures, we have $\mu_n (A^c)< \infty$ and $\mu(A^c)< \infty$.
I was trying to use a uniform integrability argument, but the definition of uniform integrability concerns a family of Random Variables. In this case, I have a family (sequence) of measures.
How to show (\ref{I})?