This problem is motivated by recent reemergence of interest in the well known identity
$$ \begin{align} \int |f|^p\,d\mu = \int^\infty_0 p\,t^{p-1}\,\mu(|f|>t)\,dt\tag{0}\label{fubini0} \end{align} $$
and a the so called cake layer integration.
I am posting this as a problem because of its multiple applications in analysis, statistics and probability.
Suppose $\nu$ is a positive Radon measure on $[0,\infty)$. Let $(X,\mathscr{F},\mu)$ be a measure space. If $f$ is a nonnegative $\mathscr{F}$-measurable function whose carrier $\{f\neq0\}$ is $\mu$ $\sigma$-finite. Show that $$ \begin{align} \int_X\nu([0,f(x))\,\mu(dx) = \int^\infty_0\mu(\{f>t\})\,\nu(dt)\tag{1}\label{fubini1} \end{align} $$
In particular, if $\phi$ is absolutely continuous on any contact subintervals of $[0,\infty)$, $\phi(0)=0$ and $\nu(dt)=\phi'(t)\,dt$,
$$ \begin{align} \int_X (\phi\circ f)\,d\mu = \int^\infty_0 \mu(\{f>t\})\,\phi'(t)\,dt\tag{3}\label{fubini2} \end{align} $$