Peano on derivative of measures: strict derivative of distributive set functions

By retracing research on coexistent magnitudes (grandeurs coexistantes) by Cauchy [9, (1841)], Peano in Applicazioni geometriche del calcolo infinitesimale [48, (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 [35, (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.