Chapter 47: Geometric measure theory
Metric outer measures; Increasing Sets Lemma; analytic spaces; inner regularity; Vitali's theorem and a density theorem; Howroyd's theorem.
Besicovitch's Density Theorem
Besicovitch's Covering Lemma; Besicovitch's Density Theorem; *a maximal theorem.
Differentiable and Lipschitz functions; smoothing by convolution; the Gagliardo-Nirenberg-Sobolev inequality; Poincaré's inequality for balls.
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.
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.
Concentration of measure
Vietoris and Fell topologies; concentration by partial reflection; concentration of measure in Rr; the Isoperimetric Theorem; concentration of measure on spheres.
Brownian motion as a stochastic process; Wiener measure on C([0,∞[)0; *as a limit of random walks; Brownian motion in Rr; invariant transformations of Wiener measure on C(]0,∞[;Rr)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.
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.
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 Rr.
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