Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?

Question

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 $$ \mu_n \rightharpoonup \mu \iff \int_X f \mathrm d \mu_n \to \int_X f \mathrm d \mu \quad \forall f \in C_c(X). $$

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

Thank you so much for your elaboration!

Answer 1

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.