If $\mu$ is a probability measure on $[0,1]$, $$f_+’(t) = \int_{(t,1]} {{s^{ - 1}}\mu (ds)}\quad \mbox{for } 0<t<1.$$ Given $\mu(\{0\})=f(0+)$, how can we calculate $$\int_{(0,1]} {{s^{ - 1}}\mu (ds)}.$$
Is it $f_+’(0)$? Here, $f_+’(0)$ can be $+\infty$.
Thanks for any suggestions;)