View source for Web differentiation

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

<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>

<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>

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" />.

The expression

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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.

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" />.

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.

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

====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>



====Comments====
In [[#References|[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 [[#References|[a1]]]. Nowadays it looks as a particular case of a theorem on convergence of martingales (cf. [[Martingale|Martingale]]) and one of the best ways to prove the [[Radon–Nikodým theorem|Radon–Nikodým theorem]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,   "Intégrales de Lebesgue. Fonctions d'ensembles. Classe de Baire" , Gauthier-Villars  (1936)</TD></TR></table>