Polar Sets and Their Applications
A polar subset of ℝ N is a set to each point of which corresponds an open neighborhood of the point that carries a superharmonic function equal to +∞ at each point of the set in the neighborhood. An inner polar set is a set whose compact subsets are polar
- PDF / 1,978,164 Bytes
- 13 Pages / 439 x 666 pts Page_size
- 58 Downloads / 261 Views
Polar Sets and Their Applications
1. Definition A polar subset of IR N is a set to each point of which corresponds an open neighborhood of the point that carries a superharmonic function equal to + 00 at each point of the set in the neighborhood. An inner polar set is a set whose compact subsets are polar. It will be shown in Section VI.2 that an analytic inner polar set is polar. If a set is (inner) polar its Kelvin transforms are also. In particular, the set of infinities of a superharmonic function is a polar subset of its domain. Conversely, it will be shown (Theorem 2) that a polar set is always a subset of the set of infinities ofa single superharmonic function defined on IR N . The polar sets are the negligible sets of classical potential theory. An assumption about points of IR N true except for the points of an [inner] polar set is said to be true [inner] quasi everywhere. A subset of an [inner] polar set is [inner] polar. A singleton g} is polar because G(e, 0) is superharmonic on IR N and equal to + 00 at e. Although the point 00 is not in IR N , that point is considered a Euclidean boundary point of every unbounded set. In a context allowing 00 in the domain of harmonic and superharmonic functions, this point is polar for N = 2 but not for N > 2. Since a superharmonic function on an open subset of IR N is IN integrable on every closed ball in its domain, and since every polar set A can be covered by a countable number of open sets, each carrying a positive superharmonic function with value + 00 on the part of A in its domain, a polar set has IN measure O. It follows that an IN measurable inner polar set also has IN measure O. If u and v are superharmonic functions on an open subset of IRN and if u = v inner quasi everywhere, or if u ~ v inner quasi everywhere, then the same relation holds IN almost everywhere and therefore [Section 11.6(0] everywhere.
J.L. Doob, Classical Potential Theory and Its Probably Counter © Springer-Verlag Berlin Heidelberg 2001
58
l.V. Polar Sets and Their Applications
2. Superharmonic Functions Associated with a Polar Set Theorem. If A is a polar subset of IR N , there is a function superharmonic on IR N and identically + 00 on A. This function can be chosen to be the potential GJ1. of a measure J1. with J1.(IR N ) finite and to be finite at any preassigned point oflR N - A.
e
To prove the theorem suppose that E IR N - A and apply the LindelOf covering theorem to cover A by balls Bo , B 1 , .•• so small that is not in any ball closure and that to each ball Bk corresponds a function Uk defined and superharmonic on an open neighborhood of lik and identically + 00 on Bk n A. Let J1.k be the projection on Bk of the Riesz measure associated with Uk; choose a strictly positive constant Ck so small that CkJ1.k(Bk) < r k, that CkIGJ1.k(e)1 < r\ and if N = 2, that Ck Ie:' log 1'71 J1.k(d'7) < r k. The superharmonic potential G:EO' CkJ1.k is + 00 on A and finite at
e
e.
Observation (a). Since the set of infinities of a superharmonic function v is the GlJ set {v > n}, every po
Data Loading...
