Web differentiation

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)