On local times for functions and stochastic processes

Let $X(t)$, $0\l t\l 1$, be a real-valued measurable function having a local time $\alpha(t,u)$, $0\l t\l 1$, $u\in\mathbb R$. If the latter is continuous in $t$ for a.a. $u$, then the distribution $F(t,x)=\int_\mathbb R\mathbb{I}\{\alpha(t,u) > x\}\,du$ and the monotone rearrangement $\alpha^*(t,u)=\inf\{x\: F(t,x) < u\}$ of the local time $\alpha(t,u)$ are the local times for $\xi(s)=\alpha(s,X(s))$ and $\xi^*(s)=F(s,\xi(s))$, $0\l s\l 1$, respectively.