• $X := \mathbb R^n$,
  • $C_b(X)$ 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!

  • $\begingroup$ 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 $C_b$. $\endgroup$
    – terceira
    Jan 12 at 21:33
  • $\begingroup$ @terceira It seems in your example $\mu_n := \delta_n$. Could you explain what is your $\mu$? $\endgroup$
    – Analyst
    Jan 12 at 21:37
  • $\begingroup$ My $\mu$ is the zero measure. $\endgroup$
    – terceira
    Jan 12 at 21:44
  • $\begingroup$ @terceira Ah my $\mu, \mu_n$ are all probability measures. $\endgroup$
    – Analyst
    Jan 12 at 21:45
  • $\begingroup$ Sorry, missed that. $\endgroup$
    – terceira
    Jan 12 at 21:47

1 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.

  • $\begingroup$ Thank you so much for your help! I have just got the same conclusion from this answer in which Stone–Weierstrass theorem is appealled. $\endgroup$
    – Analyst
    Jan 12 at 20:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.