Area formula (geometric measure theory) explained

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

f\colon R ^{n} \to R ^{m} and A\subset R ^{n} , the multiplicity function N(f,A,y), y\in R ^{m} , is the (possibly infinite) number of points in the preimage f ^{-1} (y)\cap A . The multiplicity function is also called the Banach indicatrix. Note that N(f,A,y) = l{H} ^{0(f} ^{-1} (y)\cap A) . Here, l{H} ^{n} denotes the n-dimensional Hausdorff measure, and l{L} ^{n} will denote the n-dimensional Lebesgue measure.

Theorem: If

f\colon R ^{n} \to R ^{m} is Lipschitz and n\leq m , then for any measurable A\subset R ^{n} ,$$\backslash int\_A\; (Df(x))\backslash ,\; d\; \backslash mathcal^n(x)\; =\; \backslash int\_\; N(f,A,y)\; \backslash ,\; d\backslash mathcal^n(y)\; \backslash ,,$$where$$(Df(x))=\backslash sqrt$$is the Jacobian of Df(x) .

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

The theorem was proved first by Herbert Federer .

Sources

• Book: 1857292. Ambrosio. Luigi. Fusco. Nicola. Pallara. Diego. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press. New York. 2000. 0-19-850245-1. Luigi Ambrosio. 0957.49001. Nicola Fusco.
• Book: 3409135. Evans. Lawrence C.. Gariepy. Ronald F.. Measure theory and fine properties of functions. Revised edition of 1992 original. Textbooks in Mathematics. CRC Press. Boca Raton, FL. 2015. 978-1-4822-4238-6. Lawrence Evans. 10.1201/b18333. 1310.28001.
• Book: Federer, Herbert. Herbert Federer. Geometric measure theory. Berlin–Heidelberg–New York. Springer-Verlag. Die Grundlehren der mathematischen Wissenschaften. 153. 1969. 978-3-540-60656-7. 0257325. 0176.00801 . 10.1007/978-3-642-62010-2.
• Book: 0756417. Leon Simon. Simon. Leon. Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University. 3. Australian National University, Centre for Mathematical Analysis. Canberra. 1983. 0-86784-429-9. 0546.49019.