For locally compact space $X$, Let $C_c(X)$ be the compactly supported, continuous functions on $X$ to $\mathbb{C}$. It is known theorem (Riesz-Markov-Kakutani) that the dual space of $C_c(X)$ is characterized by Radon measures on $X$, say ${\cal M}(X)$.
My question is, what is the dual space of the subspace. $$ {\cal M}_0(X) = \left\{\mu \in {\cal M}(X)\bigg| \int_X \mathrm d\mu = 0\right\} $$
Here is my initial idea: Consider an equivalence relation for $f, g \in C_c(X)$: $$ f\sim g \quad \text{if} \quad f(x)-g(x) = c \quad \text{for some }c\in\mathbb{C}, \forall x\in X $$ and we can consider equivalence class $[f]\subset C_c(X)$. The claim is the family of this equivalence class is dual to ${\cal M}_0(X)$.
The duality pairing is the natural pairing, while it is well-defined because $\mu(f) = \mu(g)$ if $f\sim g$. For metric or norm, $\|[f]\| = \inf_{f\in[f]} \|f\|$ is adopted.
How can I validate the guess, or is there other conditions that I need to refine?