Let $U:\mathbb{R}^{d} \longrightarrow [0,\infty) $ such that $\lim \limits _{|x|\rightarrow\infty} U(x) = \infty $ (i don't know if this limit property is useful for the question) and $(\nu_{n})$ a sequence of prob measures on $\mathbb{R}^{d}$.
Can I use the condition $ \sup_{n} \int\limits_{\mathbb{R}^{d}} U d\nu_{n} <\infty $ to get that for $\epsilon >0$ there exists $K\subseteq \mathbb{R}^{d}$ compact such that $\sup_{n} \int\limits_{K^{c}} U d\nu_{n} < \epsilon$ ?
Or if this doesn't hold, what would be an additional property on $U$ so that the above statement would hold ?
Finally, is this somehow related to the notion of tightness ?
I would appreciate some help because I am confused.