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2020 Mathematics Subject Classification: Primary: 49Q15 Secondary: 49Q2049Q0528A7532C3058A2558C35 [MSN][ZBL]

An area of analysis concerned with solving geometric problems via measure-theoretic techniques. The canonical motivating physical problem is probably that investigated experimentally by J. Plateau in the nineteenth century: Given a boundary wire, how does one find the (minimal) soap film which spans it? In mathematical terms: Given a boundary curve, find the surface of minimal area spanning it. (Cf. also Plateau problem.) The many different approaches to solving this problem have found utility in most areas of modern mathematics and geometric measure theory is no exception: techniques and ideas from geometric measure theory have been found useful in the study of partial differential equations, the calculus of variations, geometric analysis, harmonic analysis, and fractals.

Many variational problems (cf. also Variational calculus) are solved by enlarging the allowed class of solutions, showing that in this enlarged class a solution exists, and then showing that the solution possesses more regularity than an arbitrary element of the enlarged class. Much of the work in geometric measure theory has been directed towards placing this informal description on a formal footing appropriate for the study of surfaces.

Measure theoretic concepts

One of the central issues of geometric measure theory is to define the concepts of volume, area and length in the uttermost generality. The roots of them are obviously in measure theory. In particular, the concept of outer measure in a metric space plays a fundamental role.

Caratheodory construction

The following is a common construction of metric outer measures in metric spaces $(X, d)$.

Definition 1 If $\mathcal{C}$ is a class of subsets of $X$ containing the empty set, $\nu : \mathcal{C}\to [0, \infty]$ a set function with $\nu (\emptyset) =0$ and $\delta\in ]0, \infty]$, then for every $A\subset X$ we define \[ \mu^\delta (A) := \inf \left\{ \sum_i \nu (E_i) : \{E_i\}_{i\in \mathbb N} \subset \mathcal{C} \mbox{ covers '"`UNIQ-MathJax9-QINU`"' and '"`UNIQ-MathJax10-QINU`"'}\right\}\, \] and \[ \mu (A) := \lim_{\delta\downarrow 0}\; \mu^\delta (A)\, . \]

Observe that the latter limit exists because $\mu^\delta (A)$ is a nonincreasing function of $\delta$. This construction is often called Caratheodory construction. See Section 2.1 of [KP] (cp. also with [Fe]).

Theorem 2 Let $\nu$ and $\mu$ be as in Definition 7. Then $\mu$ is a metric outer measure.

(Cp. with Claims 1,2 and 3 in the proof of Theorem 1 in Section 2.1 of [EG]: although the reference handles the cases of Hausdorff outer measures, the proof extends verbatim to the setting above).

Hausdorff measures

The Caratheodory construction gives several generalizations of the concept of dimension and volume. The most common is due to Hausdorff (cp. with [Ha]).

Definition 3 The Hausdorff outer measure $\mathcal{H}^\alpha$ is given by such $\mu$ as in Definition 1 when we choose $\mathcal{C} = \mathcal{P} (X)$ and $\nu (A) = \omega_\alpha ({\rm diam}\, (A))^\alpha$, where \[ \omega_\alpha = \frac{\pi^{\alpha/2}}{\Gamma \left(\frac{\alpha}{2}+1\right)}\, \] (cp. with Section 2.1 of [EG]).

When $\alpha$ is a (positive) integer $n$, $\omega_n$ equals the Lebesgue measure of the unit ball in $\mathbb R^n$. With this choice the $n$-dimensional Hausdorff outer measure on the euclidean space $\mathbb R^n$ coincides with the Lebesgue measure. It must be noted, however, that some authors prefer to set the constant $\omega_\alpha$ equal to $1$ (see for instance [Ma]).

Hausdorff dimension

The following is a simple consequence of the definition (cp. with Theorem 4.7 of [Ma]).

Theorem 4 For $0\leq s<t<\infty$ and $A\subset X$ we have

• $\mathcal{H}^s (A) < \infty \Rightarrow \mathcal{H}^t (A) = 0$;
• $\mathcal{H}^t (A)>0 \Rightarrow \mathcal{H}^s (A) = \infty$.

The Hausdorff dimension ${\rm dim}_H (A)$ of a subset $A\subset X$ is then defined as

Definition 5 \begin{align*} {\rm dim}_H (A) &= \sup \{s: \mathcal{H}^s (A)> 0\} = \sup \{s: \mathcal{H}^s (A) = \infty\}\\ &=\inf \{t: \mathcal{H}^t (A) = 0\} = \inf \{t: \mathcal{H}^t (A) < \infty\}\, . \end{align*}

Other measures related to the volume

The Caratheodory construction can be used to build other generalizations of the concept of volume, such as

• The spherical $\alpha$-dimensional outer measures on a metric space $(X,d)$, see Section 2.1.2 of [KP].
• The Gross outer measures, the Caratheodory outer measures, the integral-geometric outer measures (see also Favard measure) and the Gillespie outer measures in $\mathbb R^n$, see Sections 2.1.3-2.1.4-2.1.5 of [KP] (cp. also with 2.10.2-2.10.3-2.10.4 of [Fe]).

All these measures coincide on sufficiently regular set (in particular on rectifiable sets, see below), but they are, in general, different.

Differentiation theorem

A fundamental tool in geometric measure theory is the following theorem (cp. with Differentiation of measures), usually credited to Besicovitch and which gives an explicit characterization of the Radon-Nykodim decomposition for locally finite Radon measures on the euclidean space.

Theorem 6 (cp. with Theorem 2.12 of [Ma] and Theorem 2 in Section 1.6 of [EG]) Let $\mu$ and $\nu$ be two locally finite Radon measures on $\mathbb R^n$. Then,

• the limit

\[ f(x) := \lim_{r\downarrow 0} \frac{\nu (B_r (x))}{\mu (B_r (x))} \] exists at $\mu$-a.e. $x$ and defines a $\mu$-measurable map;

• the set

\begin{equation}\label{e:singular} S:= \left\{ x: \lim_{r\downarrow 0} \frac{\nu (B_r (x))}{\mu (B_r (x))} = \infty\right\} \end{equation} is $\nu$-measurable and a $\mu$-null set;

• $\nu$ can be decomposed as $\nu_a + \nu_s$, where

\[ \nu_a (E) = \int_E f\, d\mu \] and \[ \nu_s (E) = \nu (S\cap E)\, . \] Moreover, for $\mu$-a.e. $x$ we have: \begin{equation}\label{e:Lebesgue} \lim_{r\downarrow 0} \frac{1}{\mu (B_r (x))} \int_{B_r (x)} |f(y)-f(x)|\, d\mu (y) = 0\qquad \mbox{and}\qquad \lim_{r\downarrow 0} \frac{\nu_s (B_r (x))}{\mu (B_r (x))}= 0\, . \end{equation}

Covering arguments

Theorem 6 does not hold in general metric spaces. It holds provided the metric space satisfies some properties about covering of sets with balls (cp. with Covering theorems (measure theory)). In fact, aside from their links to the differentiation of measures, both the Vitali and Besicovitch covering Lemmas (see again Covering theorems (measure theory)) and similar arguments play a pivotal role in several fundamental results of geometric measure theory.

Rectifiable sets

Also called countably rectifiable set, rectifiable sets are central objects of study in Geometric measure theory, cp. with Rectifiable set. Rectifiable sets of the euclidean space can be thought as measure-theoretic generalizations of $C^1$ submanifolds. As such they have an integer Hausdorff dimension. In the special case of $1$-dimensional sets of the euclidean space, they were first introduced by Besicovitch, as a suitable generalization of rectifiable curves. In what follows we will use the terminology $m$-dimensional rectifiable set. Some authors prefer the terminology countably $m$-rectifiable set or, briefly, $m$-rectifiable.

Rectifiable subsets of the Euclidean space $\mathbb R^n$ can be defined in several ways, see Rectifiable sets. We adopt here the following one

Definition 7 (cp. with Lemma 11.1 of [Si]) A Borel set $E\subset \mathbb R^n$ is a rectifiable subset of dimension $k$ if it has Hausdorff dimension $k$ and there is a countable family of $C^1$ $k$-dimensional submanifolds of $\mathbb R^n$ which cover $\mathcal{H}^k$-almost all $E$.

The assumption that $E$ is a Borel set might be dropped, but in this case the set might not be $\mathcal{H}^k$-measurable (see Rectifiable set). In what follows we will assume that rectifiable sets $E$ are $\mathcal{H}^k$ measurable: $\sigma$-finite $\mathcal{H}^k$-measurable sets can be decomposed into the union of a Borel set and an $\mathcal{H}^k$-null set.

Besicovitch's works

As already mentioned, the theory of rectifiable $1$-dimensional sets was first laid down by Besicovitch in a series of works (see ??-??-??). In these seminal works Besicovitch considered and solved several questions which became later pivotal for rectifiable sets of higher dimension and also introduced many useful tools (such as the Besicovitch covering theorem, see Covering theorems (measure theory)) which have been widely used in different contexts.

One-dimensional sets

The theory of one-dimensional rectifiable sets is somewhat special since much stronger theorems can be proved which fail for higher dimensions. Perhaps the most useful one is the following (cp. with Theorem 3.14 of [Fa]):

Theorem 8 A continuum, i.e. a compact connected set, $E\subset\mathbb R^n$ of finite $\mathcal{H}^1$ measure is always rectifiable and arcwise connected. Indeed it is always the image of a rectifiable curve.

We refer to [Fa] for a comprehensive account of the theory of rectifiable one-dimensional sets.

General dimension and codimension

Besicovitch-Federer projection theorem

Often, one is faced with the task of showing that some set, which is a solution to the problem under investigation, is in fact rectifiable, and hence possesses some smoothness. A major concern in geometric measure theory is finding criteria which guarantee rectifiability, several of these criteria are listed in Rectifiable set. One of the most striking results in this direction is the Besicovitch–Federer projection theorem, which illustrates the stark difference between rectifiable and unrectifiable sets. This theorem characterize purely unrectifiable $m$-dimensional sets as those sets whose projections are $\mathcal{H}^m$-negligible on almost every $m$-dimensional plane (for the precise statement, we refer to Rectifiable set). This deep result was first proved for $1$-dimensional sets in the plane by A.S. Besicovitch, and later extended to higher dimensions by H. Federer. B. White in [Wh] has shown how the higher-dimensional version of this theorem follows via an inductive argument from the planar version.

Recitifiable measures

It is also possible (and useful) to define a notion of rectifiability for (locally finite) Radon measures: A Radon measure $\mu$ is said to be $m$-rectifiable if it is absolutely continuous (cf. also Absolute continuity) with respect to the $m$-dimensional Hausdorff measure and there is an $m$-dimensional rectifiable set $E$ for which $\mu ({\mathbb R}^n\setminus E)=0$. The complementary notion of a measure $\mu$ being purely $m$-unrectifiable is defined by requiring that $\mu$ is singular with respect to all $m$-rectifiable measures (cf. also Mutually-singular measures). Thus, in particular, a Borel set $E$ is $m$-rectifiable if and only if the measure $\mu$ defnied by $\mu (A) := \mathcal{H}^m (A\cap E)$ (i.e. the restriction of $\mathcal{H}^m$ to $E$ is $m$-rectifiable; this allows one to study rectifiable sets through $m$-rectifiable measures.

Besicovitch-Marstrand-Preiss theorem

It is common in analysis to construct measures as solutions to equations, and one would like to be able to deduce something about the structure of these measures (for example, that they are rectifiable). Often, the only a priori information available is some limited metric information about the measure, perhaps how the mass of small balls grows with radius (cp. with Density of a set). Probably the strongest known result in this direction is Preiss' density theorem, which generalizes earlier results of Besicovitch and Marstrand. The following theorem summarizes both the deep results of Marstrand and Preiss (cp. with [De]).

Theorem 9 Let $\mu$ be a locally finite Radon measure in the euclidean space $\mathbb R^n$ and $\alpha$ a nonnegative real number. Then the $\alpha$-dimensional density \[ \lim_{r\downarrow 0} \frac{\mu (B_r (x))}{r^\alpha} \] exists, it is finite and positive at $\mu$-a.e. $x\in \mathbb R^n$ if and only $\alpha$ is an integer $k$ and there are a rectifiable $k$-dimensional Borel set $E$ and a Borel function $f: E\to ]0, \infty[$ such that \[ \mu (A) = \int_{A\cap E} f\, d\mathcal{H}^k\qquad \mbox{for any Borel '"`UNIQ-MathJax106-QINU`"'.} \]

Preiss' main tool in proving this result was the notion of tangent measures. A non-zero Radon measure $\nu$ is a tangent measure of $\mu$ at $x$ if there are sequences $r_i\downarrow 0$ and $c_i$ such that, for all continuous real-valued functions $\phi$ with compact support, \[ \lim_{i\to\infty} c_i \int \phi \left(\frac{y-x}{r_i}\right)\, d\mu (y) = \int \phi (y)\, d\nu (y)\, . \] Thus, an $m$-rectifiable measure will, for almost-every point, have tangent measures which are multiples of $m$-dimensional Hausdorff measure restricted to the approximate tangent plane at that point (cp. with Rectifiable set); for unrectifiable measures, the set of tangent measures will usually be much richer. The utility of the notion lies in the fact that tangent measures often possess more regularity than the original measure, thus allowing a wider range of analytical techniques to be used upon them.

References

[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fa]	K. J. Falconer. "The geometry of fractal sets". Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986. MR0867284 Zbl 0587.28004
[Fe]	H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Ha]	F. Hausdorff, "Dimension and äusseres Mass" Math. Ann. , 79 (1918) pp. 157–179 MR1511917
[KP]	S. G. Krantz, H. Parks, "Geometric Integration Theory", Birkhäuser (2008).
[Ma]	P. Mattila, "Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Mu]	M. E. Munroe, "Introduction to Measure and Integration". Addison Wesley (1953).
[Wh]	B. White, "A new proof of Federer's structure theorem for $k$-dimensional subsets of $\mathbb R^n$ J. Amer. Math. Soc. , 11 : 3 (1998) pp. 693–701