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Let $\Omega \subset \mathbb R^N$ and $f \in BV(\Omega)$. The coarea formula states that

$$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$

Unfortunately, the formula $$f = \int_{\mathbb R} \chi_{\{f >h\}} \, dh$$ does not hold. But is an analogous one true? That is, can we write in some way $f$ as integral of characteristic functions of level sets?

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    $\begingroup$ What's wrong with en.wikipedia.org/wiki/Layer_cake_representation ? $\endgroup$
    – Willie Wong
    Commented Apr 19, 2019 at 18:18
  • $\begingroup$ @WillieWong That sounds great. How can it be generalized if $f$ is not necessarily non-negative? $\endgroup$
    – Riku
    Commented Apr 19, 2019 at 18:21
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    $\begingroup$ Write $f = f_+ - f_-$? $\endgroup$
    – Willie Wong
    Commented Apr 19, 2019 at 18:22
  • $\begingroup$ @WillieWong In that case get $f(x)=\int _{0}^{\infty }1_{L(f,t)}(x)\,dt + \int _{\infty}^{0}1_{E(f,t)}(x)\,dt,$ where $L(f,t)=\{y\in {\mathbb {R}}^{n}|f_+(y)\geq t\}$ and $E(f,t)=\{y\in {\mathbb {R}}^{n}|f_-(y)\geq t\}$. Can it be written in a better way as an integral over $\mathbb R$? $\endgroup$
    – Riku
    Commented Apr 19, 2019 at 18:34
  • $\begingroup$ @Riku One could as an analogous question for complex valued functions (or for vector valued ones). So maybe you are looking for something like $f = \int_0^\infty \frac{f}{|f|} \chi_{\{|f|>t\}} \, dt$? (Of course we should define $\frac{f}{|f|}=0$ whenever $|f|=0$.) $\endgroup$
    – Skeeve
    Commented Apr 20, 2019 at 7:59

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