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A special concept in the differentiation of set functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w0972901.png" />. A web <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w0972902.png" /> is a totality of subdivisions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w0972903.png" /> of a basic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w0972904.png" /> with measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w0972905.png" /> such that
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<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/w/w097/w097290/w0972906.png" /></td> </tr></table>
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<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/w/w097/w097290/w0972907.png" /></td> </tr></table>
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A special concept in the differentiation of set functions  $  \psi $.  
 +
A web  $  N $
 +
is a totality of subdivisions  $  \{ A _ {j}  ^ {i} \} $
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of a basic space  $  X $
 +
with measure  $  \mu $
 +
such that
  
and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w0972908.png" /> it is possible to find a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w0972909.png" /> containing it. All <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729010.png" /> are measurable, and their totality approximates in a certain sense, [[#References|[1]]], all measurable sets. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729011.png" /> is fixed, the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729012.png" /> are said to be sets of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729014.png" />. For each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729015.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729016.png" /> there exists precisely one set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729017.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729018.png" /> containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729019.png" />.
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$$
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\cup _ { j } A _ {j}  ^ {i}  =  X ,
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$$
 +
 
 +
$$
 +
A _ {j _ {1}  }  ^ {i} \cap A _ {j _ {2}  }  ^ {i}  = \
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\emptyset ,\  j _ {1} \neq j _ {2} ,\  i = 1 , 2 \dots
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$$
 +
 
 +
and for each $  A _ {j _ {1}  }  ^ {i+} 1 $
 +
it is possible to find a set $  A _ {j _ {2}  }  ^ {i} $
 +
containing it. All $  A _ {j}  ^ {i} $
 +
are measurable, and their totality approximates in a certain sense, [[#References|[1]]], all measurable sets. If $  i $
 +
is fixed, the sets $  A _ {j}  ^ {i} $
 +
are said to be sets of rank $  i $.  
 +
For each point $  x _ {0} $
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and any $  n $
 +
there exists precisely one set $  A _ {n} ( x _ {0} ) $
 +
of rank $  n $
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containing the point $  x _ {0} $.
  
 
The expression
 
The expression
  
<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/w/w097/w097290/w09729020.png" /></td> </tr></table>
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$$
 +
D _ {N} ( x _ {0} )  = \lim\limits _ {n \rightarrow \infty } \
 +
 
 +
\frac{\psi [ A _ {n} ( x _ {0} ) ] }{\mu [ A _ {n} ( x _ {0} ) ] }
 +
 
 +
$$
  
is said to be the derivative of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729021.png" /> along the web <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729022.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729023.png" />, if that limit in fact exists. The concept of derived numbers along the web <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729024.png" /> can also be defined.
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is said to be the derivative of the function $  \psi $
 +
along the web $  N $
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at the point $  x _ {0} $,  
 +
if that limit in fact exists. The concept of derived numbers along the web $  N $
 +
can also be defined.
  
The simplest example of web differentiation is the differentiation of the increment of a function in one real variable by rational dyadic intervals of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729025.png" />.
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The simplest example of web differentiation is the differentiation of the increment of a function in one real variable by rational dyadic intervals of the form $  ( j / 2  ^ {i} , ( j + 1) / 2  ^ {i} ] $.
  
The web derivative of a countably-additive set function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729026.png" /> exists almost everywhere and is identical with the density of the absolutely continuous component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729027.png" />. In an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097290/w09729028.png" />-dimensional space, web differentiation of semi-open intervals whose diameters tend to zero as their ranks increase [[#References|[2]]] are usually studied.
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The web derivative of a countably-additive set function $  \psi $
 +
exists almost everywhere and is identical with the density of the absolutely continuous component of $  \psi $.  
 +
In an $  n $-
 +
dimensional space, web differentiation of semi-open intervals whose diameters tend to zero as their ranks increase [[#References|[2]]] are usually studied.
  
 
The concepts of a web and of web differentiation may be generalized to the case of abstract spaces without a measure [[#References|[3]]].
 
The concepts of a web and of web differentiation may be generalized to the case of abstract spaces without a measure [[#References|[3]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.E. Shilov,  B.L. Gurevich,  "Integral, measure and derivative: a unified approach" , Dover  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Kenyon,  A.P. Morse,  "Web derivatives"  ''Mem. Amer. Math. Soc.'' , '''132'''  (1973)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.E. Shilov,  B.L. Gurevich,  "Integral, measure and derivative: a unified approach" , Dover  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Kenyon,  A.P. Morse,  "Web derivatives"  ''Mem. Amer. Math. Soc.'' , '''132'''  (1973)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:28, 6 June 2020


A special concept in the differentiation of set functions $ \psi $. A web $ N $ is a totality of subdivisions $ \{ A _ {j} ^ {i} \} $ of a basic space $ X $ with measure $ \mu $ such that

$$ \cup _ { j } A _ {j} ^ {i} = X , $$

$$ A _ {j _ {1} } ^ {i} \cap A _ {j _ {2} } ^ {i} = \ \emptyset ,\ j _ {1} \neq j _ {2} ,\ i = 1 , 2 \dots $$

and for each $ A _ {j _ {1} } ^ {i+} 1 $ it is possible to find a set $ A _ {j _ {2} } ^ {i} $ containing it. All $ A _ {j} ^ {i} $ are measurable, and their totality approximates in a certain sense, [1], all measurable sets. If $ i $ is fixed, the sets $ A _ {j} ^ {i} $ are said to be sets of rank $ i $. For each point $ x _ {0} $ and any $ n $ there exists precisely one set $ A _ {n} ( x _ {0} ) $ of rank $ n $ containing the point $ x _ {0} $.

The expression

$$ D _ {N} ( x _ {0} ) = \lim\limits _ {n \rightarrow \infty } \ \frac{\psi [ A _ {n} ( x _ {0} ) ] }{\mu [ A _ {n} ( x _ {0} ) ] } $$

is said to be the derivative of the function $ \psi $ along the web $ N $ at the point $ x _ {0} $, if that limit in fact exists. The concept of derived numbers along the web $ N $ can also be defined.

The simplest example of web differentiation is the differentiation of the increment of a function in one real variable by rational dyadic intervals of the form $ ( j / 2 ^ {i} , ( j + 1) / 2 ^ {i} ] $.

The web derivative of a countably-additive set function $ \psi $ exists almost everywhere and is identical with the density of the absolutely continuous component of $ \psi $. In an $ n $- dimensional space, web differentiation of semi-open intervals whose diameters tend to zero as their ranks increase [2] are usually studied.

The concepts of a web and of web differentiation may be generalized to the case of abstract spaces without a measure [3].

References

[1] G.E. Shilov, B.L. Gurevich, "Integral, measure and derivative: a unified approach" , Dover (1977) (Translated from Russian)
[2] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
[3] H. Kenyon, A.P. Morse, "Web derivatives" Mem. Amer. Math. Soc. , 132 (1973)

Comments

In [1], "web differentiation" has been translated as "differentiation along a net" (Sect. 10.2). In it (Sect. 10.3), a generalization to Vitali systems is given.

The notion of web derivative for measures seems due to Ch.J. de la Vallée-Poussin [a1]. Nowadays it looks as a particular case of a theorem on convergence of martingales (cf. Martingale) and one of the best ways to prove the Radon–Nikodým theorem.

References

[a1] Ch.J. de la Vallée-Poussin, "Intégrales de Lebesgue. Fonctions d'ensembles. Classe de Baire" , Gauthier-Villars (1936)
How to Cite This Entry:
Web differentiation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Web_differentiation&oldid=17917
This article was adapted from an original article by V.A. Skvortsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article