Example

— If $X$ is a discrete space, $\mu$ a positive measure on $X$, and if one sets $\alpha(x) = \mu(\varphi_{\{x\}})$, then $\mu^*(f) = \sum_{x \in X} \alpha(x) f(x)$ for every numerical function $f \geq 0$ defined on $X$, since such a function is continuous (No. 1, Example).