Hi! Seeing Grafakos theorem 2.5.8, it is said that the finite borel measures space is dual of the space of continuous functions that tends to zero at infinity. To see this, I searched Conway for that answer to such a statement but with complex measures.
Given the above, I would like to see if the following adaptations I made were correct, with $X=\mathbb{R}^n$, measures with values at $[0,\infty[$ etc. since according to what I see, in the theorem grafakos works with finite positive measures.
Conway:
Adaptation: Let $M(\mathbb{R}^n)$ the space of all finite Borel measure with valued in $[0,\infty[$
$M(\mathbb{R}^n)$ is a vector space over $\mathbb{R}$. For $\mu\in M(\mathbb{R}^n)$, let $\|\mu\|:=\left|\mu\right|(\mathbb{R}^n)$. This is a norm on $M(\mathbb{R}^n)$.
Riesz Representation Theorem. Let $X=\mathbb{R}^n$ and $\mu\in M(\mathbb{R}^n)$, define $F_{\mu}:C_0(\mathbb{R}^n)\to [0,\infty[$ by $F_\mu(f)=\int f \, d\mu$.
Then $F_\mu\in (C_0(\mathbb{R}^n))^\ast$ and the map $\mu\mapsto F_\mu$ is an isometric isomorphism of $M(\mathbb{R}^n)$ onto $(C_0(\mathbb{R}^n))^\ast$.