Area formula (geometric measure theory)

In geometric measure theory the area formula relates the Hausdorff measure of the image of a Lipschitz map, while accounting for multiplicity, to the integral of the Jacobian of the map. It is one of the fundamental results of the field that has connections, for example, to rectifiability and Sard's theorem.

Definition: Given ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ and ${\displaystyle A\subset \mathbb {R} ^{n}}$, the multiplicity function ${\displaystyle N(f,A,y),\,y\in \mathbb {R} ^{m}}$, is the (possibly infinite) number of points in the preimage ${\displaystyle f^{-1}(y)\cap A}$.

The multiplicity function is also called the Banach indicatrix. Note that, ${\displaystyle N(f,A,y)=\mathbb {H} ^{0}(f^{-1}(y)\cap A)}$. We will denote by ${\displaystyle {\mathcal {H}}^{n}}$ the n-dimensional Hausdorff measure.

Theorem: If ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ is Lipschitz, then for any measurable ${\displaystyle A\subset \mathbb {R} ^{n}}$,

$${\displaystyle \int _{A}{J}(Df(x))\,d{\mathcal {L}}^{n}(x)=\int _{\mathbb {R} ^{m}}N(f,A,y)\,d{\mathcal {H}}^{n}(y)\,,}$$

where

$${\displaystyle {J}(Df(x))={\sqrt {\det(Df(x)^{t}Df(x))}}}$$

is the Jacobian of ${\displaystyle Df(x)}$.

The measurability of the multiplicity function is part of the claim. The Jacobian is defined almost everywhere by Rademacher's differentiability theorem.

Theorem was proved first by Herbert Federer (Federer 1969).