Mathematics – History and Overview
Scientific paper
2010-02-22
Mathematics
History and Overview
30 pages, on the occasion of the 150th anniversary of the birth of Giuseppe Peano (1858-1932)
Scientific paper
By retracing research on coexistent magnitudes (grandeurs coexistantes) by Cauchy (1841), Peano in "Applicazioni geometriche del calcolo infinitesimale" (1887) defines the "density" (strict derivative) of a "mass" (a distributive set function) with respect to a "volume" (a positive distributive set function), proves its continuity (whenever the strict derivative exists) and shows the validity of the mass-density paradigm: "mass" is recovered from "density" by integration with respect to "volume". It is remarkable that Peano's strict derivative provides a consistent mathematical ground to the concept of "infinitesimal ratio" between two magnitudes, successfully used since Kepler. In this way the classical (i.e., pre-Lebesgue) measure theory reaches a complete and definitive form in Peano's Applicazioni geometriche. A primary aim of the present paper is a detailed exposition of Peano's work of 1887 leading to the concept of strict derivative of distributive set functions and their use. Moreover, we compare Peano's work and Lebesgue's "La mesure des grandeurs" (1935): in this memoir Lebesgue, motivated by coexistent magnitudes of Cauchy, introduces a uniform-derivative of certain additive set functions, a concept that coincides with Peano's strict derivative. Intriguing questions are whether Lebesgue was aware of the contributions of Peano and which role is played by the notions of strict derivative or of uniform-derivative in today mathematical practice.
Greco Gabriele H.
Mazzucchi Sonia
Pagani Enrico M.
No associations
LandOfFree
Peano on derivative of measures, strict derivative of distributive set functions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Peano on derivative of measures, strict derivative of distributive set functions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Peano on derivative of measures, strict derivative of distributive set functions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-373523