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.

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.