Polar Sets and Capacity

Abstract

Sets on which a superharmonic function can have the value +∞ are called polar. Since superharmonic functions are locally integrable, such sets must be of Lebesgue measure zero. Indeed, polar sets are the negligible sets of potential theory and will be seen to play a role reminiscent of that played by sets of measure zero in integration. A useful result proved in Section 5.2 is that closed polar sets are removable singularities for lower-bounded superharmonic functions and for bounded harmonic functions. In Section 5.3 we will introduce the notion of reduced functions. Given a positive superharmonic function u on a Greenian open set Ω and E ⊆ Ω, we consider the collection of all non-negative superharmonic functions v on ∖ which satisfy v ≥ u on E. The infimum of this collection is called the reduced function of u relative to E in Ω. Some basic properties of reduced functions will be observed, including the fact that they are “almost” superharmonic. Later, in Section 5.7, deeper properties will be proved via an important result known as the fundamental convergence theorem of potential theory. Before that, however, we will develop the notion of the capacity of a set, beginning with compact sets. Taking u ≡ 1 and E to be compact, the above reduced function is almost everywhere equal to a potential on Ω, and the total mass of the associated Riesz measure is called the capacity of E. For arbitrary sets E, we will define inner and outer capacity and, if these are equal, will term E capacitable.