Let


* $X := \mathbb R^n$,
* $C\_b(X)$ the space of all real-valued bounded continuous,
* $C\_c(X)$ the space of all real-valued continuous functions with compact supports, and
* $C\_c^\infty(X)$ the space of all real-valued smooth functions with compact supports.


Let $\mu, \mu\_n$ be Borel probability measures on $X$. We say that $\mu\_n$ converges to $\mu$ *weakly* iff
$$
\mu\_n \rightharpoonup \mu \overset{\text{def}}{\iff} \int\_X f \mathrm d \mu\_n \to \int\_X f \mathrm d \mu \quad \forall f \in C\_b(X).
$$


Because $\mathbb R^n$ is locally compact and separable, we [have](https://math.stackexchange.com/a/4570642/1019043)
$$
\mu\_n \rightharpoonup \mu \iff \int\_X f \mathrm d \mu\_n \to \int\_X f \mathrm d \mu \quad \forall f \in C\_c(X).
$$



> 
> Can we further restrict the space of test functions to $C\_c^\infty (X)$?
> 
> 
> 


Thank you so much for your elaboration!



Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?
Any $f\in C\_c(X)$ can be uniformly approximated by functions $f\_n\in C\_c^\infty(X)$, say by convolving $f$ with appropriate mollifiers $\psi\_n\in C\_c^\infty(X)$.


So, your desired conclusion indeed follows.


