Suppose $\mu_n, \mu$ are Radon measures such that $\mu_n \to \mu$ in the sense of distributions. Does it follow that $\mu_n \to \mu$ in the sense of weak convergence? What about vague convergence?
The converse, that weak convergence implies convergence in the sense of distributions, follows directly from the definition, since for distributions our class of test functions is smaller--it only contains smooth compactly supported functions.