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On Differentiation with respect to a Function

Published online by Cambridge University Press:  24 October 2008

E. C. Francis
E. C. Francis
Affiliation:
Peterhouse
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Extract

Thirty years ago, in a paper on continued fractions, Stieltjes published a definition of the integral which bears his name. His replacement of the variable of integration x by a more general “base function” φ(x)—a change which throws so much light upon other theories of integration—received at first little attention, but has later sprung into greater prominence; so much so that Professor Hildebrandt, in summarizing these various theories in a paper to the American Mathematical Society, makes the statement that “it [the Stieltjes Integral] seems destined to play the central rôle in the integrational and summational processes of the future.” Yet even now the integral and the allied theory of differentiation with respect to a function have been subjected to little detailed analysis, and the possibilities of extension have been only touched upon. It is the object of this present paper to establish certain results which are of some value in themselves and which prepare the way for an attack upon the integral.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 6 , November 1925 , pp. 924 - 934
Copyright
Copyright © Cambridge Philosophical Society 1925

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References

* Stieltjes, T. J., “Recherches sur lea Fractions Continues,” Ann. de la Fac. des Sc. de Toulouse, vol. VIII (1894).Google Scholar

† Hildebrandt, Prof. T. H., “On Integrals related to, and extensions of, Lebesgue Integrals,” Bull. Amer. Math. Soc. vol. XXIV (1918).Google Scholar

* Denjoy, A., “Mémorie sur la Totalization des Nombre Dérivés non-sommables,” Annales de l' Ecole Normale Supérieure (1916).Google Scholar

† Ch. de la Vallée Poussin, Intégrales de Lebesgue.Google Scholar

‡ There is an ambiguity here which is usually allowed to pass unnoticed. F(x), if of bounded variation, can be split up into the difference of two increasing functions F 1(x), F 2(x), but this can be done in many ways. One definite way is indicated in all developments of the theory of functions of bounded variation and this is best regarded as standard. If we denote the corresponding functions by F 1(x), F 2(x), we can prove the following Consistency Theorem (often tacitly assumed):

If G 1(x), G 2(x) are any two increasing functions such that

and if a set E is measurable (G 1) and (G 2); then E is measurable (F 1) and (F 2) and

Since the subject is somewhat removed from the main object of the paper I omit the proof.

* Throughout this paper I will denote an interval.Google Scholar

† I.e. measurable with respect to the base function in question.Google Scholar

* Danniell, P. J., “On differentiation with regard to a function of limited variation,” Trans. Amer. Math. Soc. vol. xix (1918).Google Scholar

Danniell defines two derivatives (upper and lower) by

If has a differential coefficient equal to each. But on this definition the function which is equal to x when x ≥0, and to - x when x<0, has differential coefficient zero at the origin.

† I.e. φ; (x 1) > φ(x 2) if x 1 > x 2+φ(x+2)+if+x+1+>+x+2>Google Scholar

‡ For many purposes F (x) is “continuous with φ ” (i.e (iii) is excluded).Google Scholar

* Lebesgue, Leçons sur l' Intégration, p. 63.Google Scholar

* By this statement we mean that if one only of these inequalities is true, then the corresponding inequality of the result holds. We are not assuming D (x) bounded.Google Scholar

* This condition is made less stringent by later results in §§ 5, 6Google Scholar

* See note on p. 930.Google Scholar