Chapter 47: Geometric measure theory

471
Hausdorff measures

Metric outer measures; Increasing Sets Lemma; analytic spaces;
inner regularity; Vitali's theorem and a density theorem; Howroyd's
theorem.

472
Besicovitch's Density Theorem

Besicovitch's Covering Lemma; Besicovitch's Density Theorem; *a
maximal theorem.

473
Poincaré's inequality

Differentiable and Lipschitz functions; smoothing by convolution;
the Gagliardo-Nirenberg-Sobolev inequality;
Poincaré's inequality for balls.

474
The distributional perimeter

The divergence of a vector field; sets with locally finite
perimeter, perimeter measures and outward-normal functions; the
reduced boundary; invariance under isometries; isoperimetric
inequalities; Federer exterior normals; the Compactness Theorem.

475
The essential boundary

Essential interior, closure and boundary; the reduced boundary, the
essential boundary and perimeter measures; characterizing sets with
locally finite perimeter; the Divergence Theorem; calculating
perimeters from cross-sectional counts, and an integral-geometric
formula; Cauchy's Perimeter Theorem; the Isoperimetric Theorem for
convex sets.

476
Concentration of measure

Vietoris and Fell topologies; concentration by partial reflection;
concentration of measure in **R**^{r}; the Isoperimetric
Theorem; concentration of measure on spheres.

477
Brownian motion

Brownian motion as a stochastic process; Wiener measure on
*C*([0,∞[)_{0}; *as a limit of random walks; Brownian
motion in **R**^{r}; invariant transformations of Wiener
measure on *C*(]0,∞[;**R**^{r})_{0}; Wiener measure is
strictly positive; the strong Markov property; hitting times;
almost every Brownian path is nowhere differentiable; almost every
Brownian path has zero two-dimensional Hausdorff measure.

478
Harmonic functions

Harmonic and superharmonic functions; a maximal principle; *f* is
superharmonic iff div grad *f*≤0; the Poisson kernel and harmonic
functions with given values on a sphere; smoothing by convolution;
Brownian motion and Dynkin's formula; Brownian motion and
superharmonic functions; recurrence and divergence of Brownian
motion; harmonic measures and Dirichlet's problem; disintegrating
harmonic measures over intermediate boundaries; hitting
probabilities.

479
Newtonian capacity

Defining Newtonian capacity from Brownian hitting probabilities, and
equilibrium measures from harmonic measures; submodularity and
sequential order-continuity; extending Newtonian capacity to
Choquet-Newton capacity; Newtonian potential and energy of a Radon
measure; Riesz kernels and their Fourier transforms; energy and
(*r*-1)-potentials; alternative definitions of capacity and
equilibrium measures; analytic sets of finite capacity; polar sets;
general sets of finite capacity; Brownian hitting probabilities and
equilibrium potentials; Hausdorff measure; self-intersecting
Brownian paths; a discontinuous equilibrium potential; yet another
definition of Newtonian capacity; capacity and volume; a measure on
the set of closed subsets of **R**^{r}.

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Revised 18.2.16