Finding the inverse of a infimum function

Question

What is the inverse function of $$f:t\mapsto \inf\{s\geq 0:\mu([0,s])>t\}$$

Can I just write:

$$f^{-1}:s\mapsto \inf\{t\geq 0:\mu([0,s])\leq t\}$$

$\mu$ is a measure.

Answer 1

In general, $f$ is not bijective. If, for example, $$\mu(A)=\begin{cases}|A| & \text{ if } A \text{ is finite}\\ \infty & \text{ otherwise}\end{cases}$$

then $f(t)=0$ for all $t>0$, so an inverse function cannot exist.