Remark 2

If $G$ is metrizable, $\mathscr{B}$ can be taken to be a sequence $(V_n)$ of neighborhoods. Then $\mu$ is the limit of the sequence of measures $(\mu * f_{V_n}) \cdot \beta$ with continuous densities. If $G$ is a real Lie group, the $f_{V_n}$ can be taken to be infinitely differentiable; we shall see later on that the densities $\mu * f_{V_n}$ are then infinitely differentiable.