Thinness of a set
at a point
A local criterion for the fact that is a polar set. A non-empty set
is said to be thin at the point
in two cases:
1) is not a limit point of
; that is,
, where
is the derived set of
;
2) and there exists a superharmonic function
in a neighbourhood of
(see Superharmonic function) such that
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The set is polar if and only if it is thin at each of its points. For an arbitrary set
the subset of those points at which
is thin is polar. Any non-empty subset of a set which is thin at the point
is thin at
. The union of a finite number of sets thin at the point
is a set thin at
.
A segment in the plane is not a thin set at any of its points. If
is a thin set at a point
, then there exist arbitrarily small discs with centre
and not intersecting
. A polar set
is completely discontinuous. However, the Cantor set on the
-axis (which is of measure zero) is not thin at any of its points. At the same time, for example, in
the set of points
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which has a spine at the point , where
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is the Newton potential with density on the segment
, is a thin set at the spine
(Lebesgue's example).
References
[1] | M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1969) |
[2] | N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian) |
Comments
Two further interesting properties of thinness are: 1) is thin at
if and only if
is not a limit point of
with respect to the fine topology; and 2) a boundary point
of an open set
, bounded if
, is regular for the Dirichlet problem if and only if the complement of
is not thin at
.
The concept of thinness, and its use to define a fine topology, is fundamental in any potential theory. For example, in probabilistic potential theory associated to a strong Markov process, a Borel set is thin at
if and only if, starting from
, the process almost surely will not hit
even once. But, in general, a set thin at each of its points is not polar; a countable union of such sets is called a semi-polar set, a kind of exceptional set (related to the Dirichlet problem) which can be considerably bigger than a polar set when the potential theory lacks symmetry (for example, for the heat equation potential theory). Roughly speaking, a set
is polar (respectively semi-polar) in probabilistic potential theory if the process almost surely never meets
(respectively, only meets
at most a countable number of times). See also Potential theory, abstract.
References
[a1] | C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972) |
Thinness of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thinness_of_a_set&oldid=18916