A standard result in probability theory asserts that a sequence of probablity measures $\mu_n$ on the Borel $\sigma$-algebra of $\bf R$ converges in law or weakly to a probability measure $\mu$ if and only if the characteristic functions $\widehat{\mu_n}$ converge pointwise to $\widehat{\mu}$.
Convergence in law means that $\int f d\mu_n \rightarrow \int f d\mu$ for all $f$ bounded continuous.

When $f$ is for example $C^1$ with compact support, this follows from the formula
$$
\int_{\bf R} f(x) d\mu(x) = {1\over 2\pi} \int_{\bf R} \hat{f}(t) \hat{\mu}(t) \, dt.
$$
Unfortunately the integral on the right hand side is not well defined for all $f$ bounded continuous, so additional work is needed to get the full result.

Is there a well-known result in (Schwartz) distribution theory that can make sense of the formula for all probability measure $\mu$ and all $f$ bounded continuous so as to be able to get the convergence result directly?
$$
\langle f , \mu \rangle = {1\over 2\pi} \langle \hat{f} , \hat{\mu} \rangle.
$$
I guess that one can always build an adhoc pair of dual spaces for which this formula holds, but rarrying such construction would more or less amount to the bookkeeping done in standard probability books. Maybe convolution can help here since duality relations are sometimes particular cases of convolution relations.