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Let $f$ be a non-negative, measurable function on a $\sigma$-finite measure space $(X,\mathcal{B},\mu)$. Let $E\in\mathcal{B}$ and let \begin{equation} \phi_E(t)=\mu(\{x\in E: f(x)>t\}). \end{equation} Prove that \begin{equation} \int_E\log(1+f(x))\,\mathrm{d}\mu(x)= \int_0^\infty\frac{1}{1+t}\phi_E(t)\,\mathcal{d}t. \end{equation}

Is Fubini's theorem a viable approach? How is the condition of $\sigma$-finiteness to be used?

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    Yes, Fubini's theorem is a viable approach. The $\sigma$-finiteness ensures the applicability of the theorem. – 2017-01-02

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