I'm trying to prove this property. Could you have a check on my attempt?
Let $\mu$ be a Borel probability measure on $\mathbb R$ and $F$ its c.d.f. Then $F$ is right-continuous and non-decreasing. Let $F^{-1}$ be the quantile function of $\mu$.
Theorem: If $\mu$ is atomless then $F \circ F^{-1} (t) = t$ for all $t \in (0, 1)$.
My attempt: Let $a := F^{-1} (t)$. By construction of quantile function, there is $(x_n)$ such that $F(x_n) \ge t$ and $x_n \searrow a$. By right-continuity of $F$, $F(x_n) \searrow F(a)$ and thus $F(a) \ge t$.
Assume the contrary that $F(a) > t$. Notice that $F$ is continuous, so there is $b<a$ such that $F(b) > t$. This contradicts the minimality of $a$. This completes the proof.
