Web differentiation
A special concept in the differentiation of set functions . A web
is a totality of subdivisions
of a basic space
with measure
such that
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and for each it is possible to find a set
containing it. All
are measurable, and their totality approximates in a certain sense, [1], all measurable sets. If
is fixed, the sets
are said to be sets of rank
. For each point
and any
there exists precisely one set
of rank
containing the point
.
The expression
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is said to be the derivative of the function along the web
at the point
, if that limit in fact exists. The concept of derived numbers along the web
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 .
The web derivative of a countably-additive set function exists almost everywhere and is identical with the density of the absolutely continuous component of
. In an
-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) |
Web differentiation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Web_differentiation&oldid=49185