2.1 Measures and distributionsB288B328

B288B328

The potentials studied here are seldom differentiable. On $\mathbb R^2$, for instance, the function $\log |x|$ tends to $-\infty $ at the origin, yet its Laplacian has a precise meaning: it is $2\pi $ times the Dirac mass at the origin (proposition 3.1.13). Such statements are made rigorous by differentiating in the weak sense, that is, by moving derivatives onto smooth compactly supported test functions through integration by parts. The derivatives obtained in this way are distributions. For subharmonic functions, and later for plurisubharmonic ones, the relevant second derivatives are positive distributions, and a positive distribution is a measure. The measure attached to a potential records where its mass lies, including the mass concentrated at its singularities; measures and distributions are therefore the basic language of potential theory.

Throughout this chapter, $\Omega $ is an open subset of $\mathbb R^ d$, where $d\geq 1$. The first section introduces Radon measures and signed Radon measures and recalls, with short proofs that fix our conventions, the parts of Lebesgue integration that we use. The core of the chapter consists of two facts: the Riesz representation theorem (theorem 2.3.3), which identifies positive linear functionals on compactly supported continuous functions with Radon measures, and the regularization of distributions by convolution with a smooth kernel, which commutes with derivatives and preserves positivity (proposition 2.5.3). Together they show that positive distributions are measures (theorem 2.5.6). Since limits of potentials lead to limits of measures, we then study weak convergence and compactness of measures, for sequences and, where sequences do not suffice, for nets. The last section constructs bounded densities and proves the Lebesgue differentiation theorem; as an application, a continuous function with locally bounded distributional Hessian has a second-order Taylor expansion almost everywhere (proposition 2.8.5).

Historical remarks and references for this chapter are collected in subsection 25.1.1.