States on the $C^*$ algebra $C_0(X)$

Question

Let $X$ be a locally compact second countable space. Consider the $C^*$- algebra $C_0(X)$. Is it true that there is a one to one correspondence between states on $C_0(X)$ and Borel probability measures on $X$? Or is it true only $X$ is compact? In that case $C_0(X)=C(X)$. I have $Y$ is a compact subspace of $X$ and I have a Borel probability measure on $Y$. Can I extend it to $X$?

Answer 1

Yes, it is true. There is a one to one correspondence between states on $C_0(X)$ and Radon probability measures. This is the content of the Riesz representation theorem. However, we also have the following theorem:

If $X$ is a locally compact Hausdorff space that is second countable, then every finite Borel measure on $X$ is Radon. See Folland's "Real analysis" theorem 7.8.