On local times for functions and stochastic processes. I

Let $X(t)$, $0\le t\le 1$, be a real-valued measurable function having a local time $\alpha (t,u)$, $0\le t\le 1$, $u\in\mathbf{R}$. If the latter is continuous in $t$ for a.e. $u$, then the distribution. $F(t,x)=\int_\mathbf{R}\mathbb{I}(\alpha(t,u)>x)\,du$ 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 $\xi^*(s)=F(s,X(s))$, $0\le s\le 1$, respectively.