Let X_n in S' and mu_n, mu in M(S'). S' is the space of tempered distributions. I'm looking for a reference that says if < f, X_n > converges in distribution to < f, X> for every f in S, then mu_n weakly converges to mu where mu is the measure for the random variable X. Is anyone aware of such a result?

Also, same question for X_n in D' and mu_n, mu in M(D').

Thank you in advance for any insight, it's very much appreciated.