View source for Coarea formula

{{MSC|49Q15}}

[[Category:Classical measure theory]]

{{TEX|done}}

The coarea formula is a far-reaching generalization of [[Fubini theorem|Fubini's theorem]] in the Euclidean space using "curvilinear coordinates" or "distorted foliations". In what follows we denote by $\mathcal{H}^n$ the $n$-dimensional  [[Hausdorff measure]] in $\mathbb R^m$ and by $\lambda$ the [[Lebesgue  measure]] in any $\mathbb R^k$.

===Lipschitz maps===
Consider a Lipschitz map $f: \mathbb R^m \to \mathbb R^n$, where  $m\geq n$. Recall that, by [[Rademacher theorem|Rademacher's theorem]],  $f$ is differentiable $\lambda$-a.e.. At any point $y\in \mathbb R^n$ of  differentiability we denote by $J f (y)$ the [[Jacobian]] of $f$ in  $y$, that is the square root of the determinant of $Df|_y^t \cdot Df|_y$  (which, by the [[Cauchy Binet formula]], equals the sum of the squares  of the determinants of all $n\times n$
minors of the [[Jacobian|Jacobian matrix]] $Df|_y$, see [[Jacobian]]). 

'''Theorem 1'''
Let $f: \mathbb R^m \to \mathbb R^n$ be a Lipschitz function. Then for each $\lambda$-measurable $A\subset \mathbb R^m$, we have

(a) The map $y\mapsto J f (y)$ is Lebesgue measurable;

(b) The set $A\cap f^{-1} (z)$ is an $m-n$-dimensional [[Rectifiable set|rectifiable set]] for $\lambda$-a.e. $z\in \mathbb R^n$;

(c) The map
\[
z\mapsto \mathcal{H}^{m-n} (A\cap f^{-1} (\{z\}))
\]
is Lebesgue measurable;

(d) The following formula holds
\begin{equation}\label{e:coarea}
\int_A Jf (y)\, dy = \int_{\mathbb R^n} \mathcal{H}^{m-n} (A\cap f^{-1} (\{z\})\, dz\, .
\end{equation}

Cp. with Theorem 1 of Section 3.4.2 of {{Cite|EG}} concerning the points (a), (c) and (d). For point (a) see Theorem 2.93 of {{Cite|AFP}}. The statement of Theorem 1 can be considerably generalized: in particular one can consider Lipschitz maps $f: A \to E$ where $A$ is an $m$-dimensional rectifiable subset of $\mathbb R^M$ and $E$ is an $n$-dimensional rectifiable subset of $\mathbb R^N$. In this case the Jacobian must be suitably defined using an appropriate concept of tangential differentiation. We refer the reader to Theorem 3.2.22 of {{Cite|Fe}}.
====Fubini type statement====
A relatively simple corollary of Theorem 1 is given by the following more general statement, which is also often referred to as Coarea formula.

'''Theorem 2'''
Let $f$ be as in Theorem 1 and $g: \mathbb R^m\to \mathbb R$ a $\lambda$-summable function.
Then the map $g|_{f^{-1} \{z\}}$ is $\mathcal{H}^{m-n}$ summable for $\lambda$-a.e. $z\in \mathbb R^n$ and the following formula holds
\begin{equation}\label{e:coarea2}
\int_{\mathbb R^m} g (y)\, Jf (y)\, dy = \int_{\mathbb R^n} \int_{f^{-1} (\{z\})} g(w)\, d\mathcal{H}^{m-n} (w)\, dz\, .
\end{equation}

Cp. with Theorem 2 of Section 3.4.3 of {{Cite|EG}}. Observe that when $f$ is the projection onto the first $n$ coordinates, \eqref{e:coarea2} reduces to the classical [[Fubini theorem]]. 
====Relation to Sard's theorem====
Assume $f$ in Theorem 1 is of class $C^r$ for some $r> m-n$. Then by [[Sard theorem|Sard's theorem]] we conclude that $\lambda$-a.e. $z\in \mathbb R^n$ is a regular value of $f$ and hence that $f^{-1} (z)$ is a $C^r$ $m-n$-dimensional submanifold of $\mathbb R^m$. Since $C^1$ submanifolds are rectifiable sets, this implies conclusion (b) in Theorem 1. Thus Theorem 1 can also be considered as an appropriate generalization of Sard's theorem.
===BV functions===
In case $n=1$ the Lipschitz regularity of $f$ can be considerably relaxed leading to the Fleming-Rishel coarea formula for [[Function of bounded variation|functions of bounded variation]]: compare to the Section ''Coarea formula'' therein.

===References===
{|
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|valign="top"|{{Ref|EG}}||   L.C. Evans, R.F. Gariepy,  "Measure theory   and fine properties of   functions" Studies in Advanced  Mathematics.  CRC  Press, Boca Raton,   FL,  1992. {{MR|1158660}}  {{ZBL|0804.2800}} 
|-
|valign="top"|{{Ref|Fe}}|| H. Federer, "Geometric measure theory", Springer-Verlag (1979). {{MR|0257325}} {{ZBL|0874.49001}}
|-
|valign="top"|{{Ref|Ma}}||       P. Mattila, "Geometry of sets and measures in euclidean spaces".       Cambridge Studies in Advanced Mathematics, 44. Cambridge University       Press, Cambridge,  1995. {{MR|1333890}} {{ZBL|0911.28005}}
|-
|valign="top"|{{Ref|Si}}||   L. Simon, "Lectures on geometric measure theory", Proceedings of the   Centre for Mathematical Analysis, 3. Australian National University.   Canberra (1983) {{MR|0756417}} {{ZBL|0546.49019}} 
|-
|}