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

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!

• You seem to be missing some sort of tightness condition. Suppose that $\mu_n$ is the point measure at $\{n\}$. Then this converges to zero along $C_c$ but not along $C_b$. Jan 12 at 21:33
• @terceira It seems in your example $\mu_n := \delta_n$. Could you explain what is your $\mu$? Jan 12 at 21:37
• My $\mu$ is the zero measure. Jan 12 at 21:44
• @terceira Ah my $\mu, \mu_n$ are all probability measures. Jan 12 at 21:45
• Sorry, missed that. Jan 12 at 21:47

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 $$\psi_n\in C_c^\infty(X)$$.