uniformly continuous functions and measures Consider  a sequence of measures for which $\int f \, \mathrm{d}\mu_{n} \to \int f \,  \mathrm{d}\mu$ for all uniformly continuous functions $f$. Is it true that $\mu_n \to \mu$ strongly?

Consider $\mu_n$ an approximation to the identity; e.g. $$\mu_n = \frac n 2 \chi_{[-1/n, 1/n]} \lambda$$ for $\lambda$ the Lebesgue measure. Then $\int f\ d\mu_n \to \int f\ d\delta = f(0)$ for all continuous $f$; but $\mu_n({0})=0\to0\ne1=\delta({0})$.
