Let $G$ be a locally compact abelian group and $M(G)$ the Banach space of all complex Radon measures on $G$. The convolution on $M(G)$ is defined by specifying a linear functional on $C_0(G)$ (and invoking Riesz representation theorem) as follows: $$\int_G\phi\,\mathrm{d}(\mu*\nu)=\int_G\int_G\phi(xy)\,\mathrm{d}\mu(x)\,\mathrm{d}\nu(y),\text{ for }\phi\in C_0(G).$$
My question is:
Why does it continue to hold for $\phi\in C_b(G)$, the space of bounded continuous functions on $G$?
I don't know how to approximate $C_b(G)$ from $C_0(G)$... Thanks in advance!
Edit: The same equality for $C_b(G)$ is needed to establish that $\widehat{\mu*\nu}=\widehat{\mu}*\widehat{\nu}$, where $\phi\in\widehat{G}$.
Note that complex measures are by definition finite.