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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t0926301.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t0926302.png" />''
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A local criterion for the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t0926303.png" /> is a [[Polar set|polar set]]. A non-empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t0926304.png" /> is said to be thin at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t0926305.png" /> in two cases:
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{{TEX|auto}}
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{{TEX|done}}
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t0926306.png" /> is not a limit point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t0926307.png" />; that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t0926308.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t0926309.png" /> is the [[Derived set|derived set]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263010.png" />;
+
'' $  E \subset  \mathbf R  ^ {n} $
 +
at a point $  y _ {0} \in \mathbf R  ^ {n} $''
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263011.png" /> and there exists a superharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263012.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263013.png" /> (see [[Superharmonic function|Superharmonic function]]) such that
+
A local criterion for the fact that  $  E $
 +
is a [[Polar set|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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263014.png" /></td> </tr></table>
+
1)  $  y _ {0} $
 +
is not a limit point of  $  E $;  
 +
that is,  $  y _ {0} \notin E  ^  \prime  $,
 +
where  $  E  ^  \prime  $
 +
is the [[Derived set|derived set]] of  $  E $;
  
The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263015.png" /> is polar if and only if it is thin at each of its points. For an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263016.png" /> the subset of those points at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263017.png" /> is thin is polar. Any non-empty subset of a set which is thin at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263018.png" /> is thin at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263019.png" />. The union of a finite number of sets thin at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263020.png" /> is a set thin at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263021.png" />.
+
2)  $  y _ {0} \in E  ^  \prime  $
 +
and there exists a superharmonic function  $  v ( x) $
 +
in a neighbourhood of $  y _ {0} $(
 +
see [[Superharmonic function|Superharmonic function]]) such that
  
A segment in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263022.png" /> is not a thin set at any of its points. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263023.png" /> is a thin set at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263024.png" />, then there exist arbitrarily small discs with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263025.png" /> and not intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263026.png" />. A polar set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263027.png" /> is completely discontinuous. However, the Cantor set on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263028.png" />-axis (which is of measure zero) is not thin at any of its points. At the same time, for example, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263029.png" /> the set of points
+
$$
 +
\lim\limits _ {\begin{array}{c}
 +
x \rightarrow y _ {0} \\
 +
x \in E \setminus  \{ y _ {0} \}
 +
\end{array}
 +
} \
 +
\inf  v ( x)  > v ( y _ {0} ).
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263030.png" /></td> </tr></table>
+
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} $.
  
which has a spine at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263031.png" />, where
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263032.png" /></td> </tr></table>
+
$$
 +
= \{ {( x, y, z) } : {V ( x, y, z) \geq  k > 1 } \}
 +
,
 +
$$
  
is the [[Newton potential|Newton potential]] with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263033.png" /> on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263034.png" />, is a thin set at the spine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263035.png" /> (Lebesgue's example).
+
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|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Landkof,  "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Landkof,  "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Two further interesting properties of thinness are: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263036.png" /> is thin at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263037.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263038.png" /> is not a limit point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263039.png" /> with respect to the [[Fine topology|fine topology]]; and 2) a boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263040.png" /> of an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263041.png" />, bounded if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263042.png" />, is regular for the Dirichlet problem if and only if the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263043.png" /> is not thin at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263044.png" />.
+
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|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|fine topology]], is fundamental in any potential theory. For example, in probabilistic potential theory associated to a strong [[Markov process|Markov process]], a Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263045.png" /> is thin at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263046.png" /> if and only if, starting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263047.png" />, the process almost surely will not hit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263048.png" /> 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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263049.png" /> is polar (respectively semi-polar) in probabilistic potential theory if the process almost surely never meets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263050.png" /> (respectively, only meets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092630/t09263051.png" /> at most a countable number of times). See also [[Potential theory, abstract|Potential theory, abstract]].
+
The concept of thinness, and its use to define a [[Fine topology|fine topology]], is fundamental in any potential theory. For example, in probabilistic potential theory associated to a strong [[Markov process|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|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|Potential theory, abstract]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Constantinescu,  A. Cornea,  "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Constantinescu,  A. Cornea,  "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR></table>

Latest revision as of 08:25, 6 June 2020


$ 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)

Comments

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.

References

[a1] C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972)
How to Cite This Entry:
Thinness of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thinness_of_a_set&oldid=48966
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article