In the book Optimal Transport for Applied Mathematicians, the author states that if $X$ is separable and locally compact, then the space of finite signed measures $\mathcal M(X)$ is dual to $C_0(X)$. Then, he goes on to claim that one can also establish a duality with $C_b(X)$. He then says that only if $X$ is compact one can claim that $C_0(X) = C_b(X)$.
My question is then, when $X$ is not compact, how come $\mathcal M(X)$ is dual with both $C_0(X)$ and $C_b(X)$? What am I missing?