Would it be true that $\mu_n \to \mu$ strongly if $\int f\mathrm{d}\mu_{n}\to \int f\mathrm{d}\mu$ for every uniformly continuous function? Assume the space is $\mathbb{R}^{N}$ and usual topology. I am reading some paper and it seems to use this as a fact. I cannot see the difference between assuming $\mu_n \Rightarrow \mu$ and this condition mentioned above(uniformly continuous function). $\Rightarrow$ (weak convergence) obviously does not imply strong convergence, and so I am confused. The paper which i am talking about is "Weak convergence of a sequence of Markov Chains" - AF Karr, it is not a HW problem.

Would it be true that $\mu_n \to \mu$ strongly if $\int f\mathrm{d}\mu_{n}\to \int f\mathrm{d}\mu$ for every uniformly continuous function? Assume the space is $\mathbb{R}^{N}$ and has the usual topology. I am reading some paper and it seems to use this as a fact. I cannot see the difference between assuming $\mu_n \Rightarrow \mu$ and this condition mentioned above  (uniformly continuous function). $\Rightarrow$ (weak convergence) obviously does not imply strong convergence, and so I am confused. The paper which I am talking about is "Weak convergence of a sequence of Markov Chains" - AF Karr, it is not a HW problem.