I have come across this question : Can I write the integral of a function in terms of its level sets? and it makes me wonder if the L1 distance between $f$ and $g$, two bounded function over a compact (let say $[0,1]^{n}$ for simplicity) can be written in terms of the Lebesgue measure (denoted $L$) of the symmetric difference between $f^{-1}(]-\infty,\lambda[)$ and $g^{-1}(]-\infty,\lambda[)$.
More precisely, does the following expression holds ? :
$$\int_{[0,1]^{n}}|f(x)-g(x)|dx=\int_{-\infty}^{\infty}L(f^{-1}(]-\infty,\lambda[)\Delta g^{-1}(]-\infty,\lambda[))d\lambda$$
I think that i have a proof using some Fubini's theorems (as we work on a compact and $f$ and $g$ are bounded) but may be I have made a mistake manipulating integrals without enough caution and it require stronger conditions.
Also, I wonder if the compactness assumption and the "boundedness" assumption are necessary ?