Suppose $\mathcal{M}$ is a set of probability measures. What does it mean to say that a mapping $\mu \rightarrow \int_{\mathbb{R}}f \ d\mu$ is continuous on $\mathcal{M}$? How could one prove that this mapping is continuous?
Would it suffice to show: $\int_{\mathbb{R} \setminus [-a,a]} f \ d\mu$ converges to $0$ uniformly in $\mu$ as $n \rightarrow \infty$?
Any sources on this would be appreciated.