Dual of $C_0(X)$, space of all continuous functions vanishing at infinity

Question

Let $X$ be locally compact, Hausdorff space. We know that the dual of $C_0(X)$( continuous functions vanishing at infinity) is the set of all complex regular Borel measure on $X$.

Now consider $X= \mathbb{N}$ with usual topology. Then $X$ is locally compact, Hausdorff. Let us consider N with counting measure. Then $C_0(\mathbb{N})$ is the space of all sequences converging to 0, denoted by $c_0$. We know that the dual of $c_0$ is $l_1$( space of all summable sequences).

My question: (1)Are there any other special type of space for which dual of $C_0(X)$ will be $L_1(X)$? (2) Is the above true for $ \mathbb{R}$ with usual topology and Lebesgue measure?

Answer 1

Suppose $\mu$ is a (regular Borel) measure such that $L^1(\mu)=C_0(X)^*$, i.e. for every regular Borel measure $\nu$ on $X$ we have a function $f\in L^1(\mu)$ such that for every Borel $A\subseteq X$ we have $\nu(A)=\int_Af(x)\,\mathrm{d}\mu(x)$.

This implies that in particular, this holds for all Dirac measures $\delta_x$ for $x\in X$, which implies that for each $x\in X$ we must have $\mu(\{x\})>0$ and $\mu(\{x\})<\infty$.

Since $\mu(\{x\})<\infty$, by regularity, we conclude that the space $X$ must be locally countable: if every neighbourhood of a point was uncountable, then they would all have infinite measure, and so by regularity, so would $\{x\}$.

I suspect that if $X$ is locally compact and locally countable, then you can find such a measure, though I don't see any obvious construction in general.