Let a metric space, the algebra of its Borel sets, a sequence of probability measures on and the space of bounded continuous functions on .
We say the sequence converges weakly (e.g. in distribution) to the probability measure if, for each :
Now suppose that, given a sequence of measures on , we have:
for every infinitely differentiable function with compact support.
How to you prove this implies that converges weakly to ?
The Weierstrass theorem only applies to continuous functions on a compact, so it can't be used here. I wouldn't be surprised if this is a classic, but I would greatly appreciate some help.