# On Radon measures with values in Banach space

It is known that continuous linear functionals on the space $C_0({\mathbb{R}^n})$ are bounded Radon measures ${\cal M}({\mathbb{R}^n})$ where $C_0({\mathbb{R}^n})$ is uniform closure of the space of continuous functions with compact support $C_c({\mathbb{R}^n})$ (for reference e.g. Folland's book on real analysis chapter 7).

I am wondering if there is a similar more general result, namely: what would be a continuous linear function on $C_0({\mathbb{R}^n})$ with values in some Banach space $E$? In analogy with the previous case where we had functionals, I assume it might be in the space of vector-valued bounded Radon measures ${\cal M}({\mathbb{R}^n};E)$?

• I suggest to look at the Radon-Nikodym property. Dec 2, 2013 at 15:25
• If your functions have values in $E$, then you expect the measures to have values in $E^\ast$ at least. If $E^\ast$ is separable, then it has the Radon-Nikodym Property. Dec 2, 2013 at 18:55

• I have had $( C_0(\mathbb{R}^n;E) )'$ on my mind: is it equal to ${\cal M}(\mathbb{R}^n;E)$? From your post I conclude it should be: $( C_0(\mathbb{R}^n;E) )'={\cal M}(\mathbb{R}^n;E')$. Thank you for your answer and the given reference book, I'll check it. Dec 3, 2013 at 13:00