Complete monotonicity and $\sigma$-finiteness

Let $f:(0, \infty) \rightarrow (0, \infty)$ be a completely monotone function such that $f(0+)=\infty$, and by Bersntein Thoerem let $\mu$ be a positive measure on $[0, \infty)$ such that

$$f(x)=\int_{[0, \infty) } e^{-t x}\mu(dt), \qquad x>0. $$

Then necessarily $\mu$ is not a finite measure. All examples of $\mu$ I can think of are however $\sigma$-finite.

Is there an example of $f$ for which $\mu$ is not $\sigma$-finite? Or is $\sigma$-finiteness a result of the construction of $\mu$ in Bernstein Theorem (which I suspect)?