# Thinness of a set

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$E \subset \mathbf R ^ {n}$ at a point $y _ {0} \in \mathbf R ^ {n}$

A local criterion for the fact that $E$ is a polar set. A non-empty set $E \subset \mathbf R ^ {n}$ is said to be thin at the point $y _ {0} \in \mathbf R ^ {n}$ in two cases:

1) $y _ {0}$ is not a limit point of $E$; that is, $y _ {0} \notin E ^ \prime$, where $E ^ \prime$ is the derived set of $E$;

2) $y _ {0} \in E ^ \prime$ and there exists a superharmonic function $v ( x)$ in a neighbourhood of $y _ {0}$( see Superharmonic function) such that

$$\lim\limits _ {\begin{array}{c} x \rightarrow y _ {0} \\ x \in E \setminus \{ y _ {0} \} \end{array} } \ \inf v ( x) > v ( y _ {0} ).$$

The set $E$ is polar if and only if it is thin at each of its points. For an arbitrary set $E$ the subset of those points at which $E$ is thin is polar. Any non-empty subset of a set which is thin at the point $y _ {0} \in \mathbf R ^ {n}$ is thin at $y _ {0}$. The union of a finite number of sets thin at the point $y _ {0} \in \mathbf R ^ {n}$ is a set thin at $y _ {0}$.

A segment in the plane $\mathbf R ^ {2}$ is not a thin set at any of its points. If $E \subset \mathbf R ^ {2}$ is a thin set at a point $y _ {0}$, then there exist arbitrarily small discs with centre $y _ {0}$ and not intersecting $E$. A polar set $E \subset \mathbf R ^ {2}$ is completely discontinuous. However, the Cantor set on the $x$- axis (which is of measure zero) is not thin at any of its points. At the same time, for example, in $\mathbf R ^ {3}$ the set of points

$$E = \{ {( x, y, z) } : {V ( x, y, z) \geq k > 1 } \} ,$$

which has a spine at the point $( 0, 0, 0)$, where

$$V ( x, y, z) = \ \int\limits _ { 0 } ^ { 1 } \frac{t dt }{\sqrt {( x - t) ^ {2} + y ^ {2} + z ^ {2} } }$$

is the Newton potential with density $t$ on the segment $( 0 \leq x \leq 1, 0, 0)$, is a thin set at the spine $( 0, 0, 0) \in E ^ \prime$( 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)

Two further interesting properties of thinness are: 1) $E$ is thin at $x$ if and only if $x$ is not a limit point of $E$ with respect to the fine topology; and 2) a boundary point $x$ of an open set $U$, bounded if $U \subset \mathbf R ^ {2}$, is regular for the Dirichlet problem if and only if the complement of $U$ is not thin at $x$.
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 $E$ is thin at $x$ if and only if, starting from $x$, the process almost surely will not hit $E$ 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 $E$ is polar (respectively semi-polar) in probabilistic potential theory if the process almost surely never meets $E$( respectively, only meets $E$ at most a countable number of times). See also Potential theory, abstract.