= Solution
For every $t\geq0$, the fundamental theorem of calculus gives
$$
\varphi(t)=\int_0^t\varphi'(s)\,ds
=\int_0^\infty\varphi'(s)\mathbf1_{\{s<t\}}\,ds.
$$
Apply this with $t=|F(x)|$. Since the integrand is nonnegative, <Tonelli theorem> permits exchanging the integrals:
$$
\begin{aligned}
\int_X\varphi(|F(x)|)\,d\mu(x)
&=\int_X\int_0^\infty
\varphi'(s)\mathbf1_{\{s<|F(x)|\}}\,ds\,d\mu(x)\\
&=\int_0^\infty\varphi'(s)
\mu(\{|F|>s\})\,ds.
\end{aligned}
$$
This is the generalized <layer cake representation>.
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