Geometric Measure Theory: Foundations¶

Job: build Hausdorff measure, rectifiability, BV / finite perimeter, and area–coarea as the measure-theoretic substrate for geometric variational problems.

Core¶

Hausdorff measure¶

For $E\subset\mathbb{R}^n$, $\delta>0$, $s\ge 0$, $$ \mathcal{H}^s_\delta(E)=\inf\Bigl{\sum_i\omega_s\Bigl(\frac{\operatorname{diam}U_i}{2}\Bigr)^s:E\subset\bigcup U_i,\;\operatorname{diam}U_i\le\delta\Bigr}, $$ $\mathcal{H}^s(E)=\lim_{\delta\downarrow 0}\mathcal{H}^s_\delta(E)$. Hausdorff dimension: $$ \dim_{\mathcal{H}}E=\inf{s:\mathcal{H}^s(E)=0}=\sup{s:\mathcal{H}^s(E)=\infty}. $$ $\mathcal{H}^n$ coincides with Lebesgue measure $\mathcal{L}^n$ up to normalization on $\mathbb{R}^n$. Lipschitz maps increase $\mathcal{H}^s$ by at most $\mathrm{Lip}^s$.

Rectifiability¶

$E\subset\mathbb{R}^n$ is countably $\mathcal{H}^k$-rectifiable if $E\subset Z\cup\bigcup_{i=1}^\infty f_i(\mathbb{R}^k)$ with $\mathcal{H}^k(Z)=0$ and $f_i$ Lipschitz. Equivalently (Besicovitch–Federer): approximate tangent $k$-planes exist $\mathcal{H}^k$-a.e. on $E$.

Purely unrectifiable: intersects every Lipschitz image of $\mathbb{R}^k$ in an $\mathcal{H}^k$-null set. Structure theorem: any $\mathcal{H}^k$-measurable $E$ with $\mathcal{H}^k(E)<\infty$ splits into rectifiable + purely unrectifiable parts.

Approximate tangent. $T_xE$ exists if rescaled sets $\eta_{x,r}(E)$ converge in measure to a $k$-plane as $r\downarrow 0$. Rectifiable sets behave like $C^1$ submanifolds measure-theoretically.

Functions of bounded variation¶

$u\in L^1_{\mathrm{loc}}(\Omega)$, $\Omega\subset\mathbb{R}^n$ open, is in $\mathrm{BV}(\Omega)$ if the distributional gradient $Du$ is a finite Radon measure: $$ |Du|(\Omega)=\sup\Bigl{\int_\Omega u\operatorname{div}\varphi:\varphi\in C_c^1(\Omega;\mathbb{R}^n),\;|\varphi|\le 1\Bigr}<\infty. $$ $\mathrm{BV}\hookrightarrow L^{1^*}(\Omega)$ (Sobolev-type) on nice domains; compact $L^1_{\mathrm{loc}}$ embedding. Structure: $$ Du=\nabla u\,\mathcal{L}^n + D^j u + D^c u $$ (approximate gradient, jump part on rectifiable $\mathcal{H}^{n-1}$ set, Cantor part).

Sets of finite perimeter¶

$E\subset\mathbb{R}^n$ has finite perimeter in $\Omega$ if $\chi_E\in\mathrm{BV}(\Omega)$. Perimeter $P(E;\Omega)=|D\chi_E|(\Omega)$.

Reduced boundary $\partial^*E$: points where a measure-theoretic outer unit normal $\nu_E(x)$ exists and $$ \lim_{r\downarrow 0}\frac{D\chi_E(B_r(x))}{|D\chi_E|(B_r(x))}=\nu_E(x). $$ De Giorgi structure theorem. $\partial^*E$ is countably $\mathcal{H}^{n-1}$-rectifiable, $|D\chi_E|=\mathcal{H}^{n-1}\llcorner\partial^*E$, and $\nu_E$ is the approximate normal. Gauss–Green: $$ \int_E\operatorname{div}\varphi=\int_{\partial^*E}\varphi\cdot\nu_E\,d\mathcal{H}^{n-1}. $$

Area and coarea formulae¶

Area formula. $f\in\mathrm{Lip}(\mathbb{R}^k,\mathbb{R}^n)$, $k\le n$, $g\ge 0$ measurable: $$ \int_{\mathbb{R}^k}g(x)\,J_k f(x)\,dx=\int_{\mathbb{R}^n}\sum_{x\in f^{-1}(y)}g(x)\,d\mathcal{H}^k(y), $$ $J_kf=\sqrt{\det(Df^*Df)}$ (or $0$ where not differentiable — Rademacher).

Coarea formula. $f\in\mathrm{Lip}(\mathbb{R}^n,\mathbb{R}^m)$, $n\ge m$, $g\ge 0$: $$ \int_{\mathbb{R}^n}g\,J_m f\,dx=\int_{\mathbb{R}^m}\int_{f^{-1}(y)}g\,d\mathcal{H}^{n-m}\,dy. $$ For $m=1$: $\int g\,|Df|=\int_{-\infty}^\infty\int_{f^{-1}(t)}g\,d\mathcal{H}^{n-1}\,dt$. Fundamental for slicing BV functions and perimeters of superlevel sets: $$ |Du|(\Omega)=\int_{-\infty}^\infty P({u>t};\Omega)\,dt\quad(u\in\mathrm{BV}). $$

Currents and varifolds (pointers)¶

Currents. An $k$-current on $\mathbb{R}^n$ is a continuous linear functional on compactly supported smooth $k$-forms. Mass $\mathbf{M}(T)$ generalizes volume; boundary $\partial T(\omega)=T(d\omega)$. Integer rectifiable currents generalize oriented rectifiable sets with integer multiplicity; Federer–Fleming: compactness and closure under mass bounds + boundary mass bounds.

Varifolds. Radon measures on the Grassmannian bundle $\mathbb{R}^n\times\mathrm{Gr}(k,n)$ — unoriented $k$-plane fields. Stationary varifolds: first variation zero (weak mean curvature zero). Allard regularity: stationary integral varifolds with small tilt excess are $C^{1,\alpha}$ graphical.

See it¶

Isoperimetric inequality via GMT. Among sets of finite perimeter with fixed $|E|$, balls minimize $P(E)$. Modern proofs: Schwarz symmetrization, or Gagliardo–Nirenberg–Sobolev on $\mathrm{BV}$ ($\|\chi_E\|_{L^{n/(n-1)}}\le C|D\chi_E|$). Equality cases characterized (up to null sets) by balls.

Jump set of $\mathrm{BV}$. For $u\in\mathrm{BV}$, the jump set $J_u$ is countably $\mathcal{H}^{n-1}$-rectifiable; $D^ju=(u^+-u^-)\nu_u\,\mathcal{H}^{n-1}\llcorner J_u$. Model: $u=\chi_E$ recovers the reduced boundary.

Check¶

Why does Rademacher’s theorem matter for the area formula?

Lipschitz maps are differentiable $\mathcal{L}^k$-a.e.; the Jacobian $J_kf$ is defined a.e. and the change-of-variables / covering arguments go through. Without a.e. differentiability, the approximate Jacobian calculus fails in this form.

If $E$ is open with smooth boundary, identify $\partial^*E$ and $P(E)$.

$\partial^*E=\partial E$ up to $\mathcal{H}^{n-1}$-null sets; $P(E)=\mathcal{H}^{n-1}(\partial E)$; $\nu_E$ is the classical outer normal. Finite perimeter strictly generalizes smooth domains (and Lipschitz, Caccioppoli, …).

Pitfalls¶

$\mathcal{H}^{n-1}(\partial E)$ vs perimeter: topological boundary can be huge (fat Cantor constructions); perimeter sees only the reduced/measure-theoretic boundary.
Cancellation: currents of opposite orientation can cancel mass in weak limits; varifolds do not cancel — choose the category to match the variational problem.
Dimension of singularities depends on codimension and optimality notion (mass-minimizing current vs stationary varifold vs mean curvature flow).
Approximate vs classical tangents: GMT statements are a.e.; pointwise everywhere geometry may fail on dense null sets.

Refs¶

Maggi, Sets of Finite Perimeter and Geometric Variational Problems
Simon, Lectures on Geometric Measure Theory
Ambrosio–Fusco–Pallara, Functions of Bounded Variation and Free Discontinuity Problems
Mattila, Geometry of Sets and Measures in Euclidean Spaces
Federer, Geometric Measure Theory

Next¶

GMT overview · Minimal surfaces · Measure theory · Differential geometry · Functional analysis