I am looking at the following theorem (16) on pg. 22. of this paper:
"Let $(\Omega,\mathcal{A},\mu)$ be a finite measure space, $f$ a non-negative, real-valued measurable function, and $\varphi:[0,\infty) \to [0,\infty)$ a continuously differentiable and monotonically increasing function with $\varphi(0)=0$. Then
\begin{equation} \int \varphi \circ f \, d\mu \; = \; \int \limits_{0}^{\infty} \varphi'(x) \, \mu(f>x) \, dx. \label{equ_mt} \end{equation} "
I currently have an integral of this form where $\varphi$ is either positive and monotone decreasing, or negative and monotone increasing. So $\varphi$ in my case fulfills either only one of the two conditions for the above theorem.
Can someone provide a modified version of the above theorem, where $\varphi$ is positive and monotone decreasing, or negative and monotone increasing (e.g. maybe set "$f<x$" in the right part of the equation)?