The Layer-Cake representation of a non-negative measureable function $f:\mathbb{R}^n\longrightarrow \mathbb{R}$ is given by $$f(x) = \int^{\infty}_{0} \mathbb{I}_{\{y\ \in\ \mathbb{R}^n|f(y)>t\}}(x)\ dt$$ Can this be generalized to functions that are not necessarily non-negative. For a non-positive function $-f$ will do. But what about the others?
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For measurable $f$, I think that
$$f(x) = \int^{\infty}_{0} \mathbb{I}_{\{y\ \in\ \mathbb{R}^n|f(y)>t\}}(x)\ dt - \int^{\infty}_{0} \mathbb{I}_{\{y\ \in\ \mathbb{R}^n|f(y)<-t\}}(x)\ dt$$
should work.
John M
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