# Metric space valued functions of bounded variation

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## Abstract

Metric space valued functions of bounded variation ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze LUIGIAMBROSIO Metric space valued functions of bounded variation Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 17, no 3 (1990), p. 439-478. <http://www.numdam.org/item?id=ASNSP_1990_4_17_3_439_0> © Scuola Normale Superiore, Pisa, 1990, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Metric Space Valued Functions of Bounded Variation LUIGI AMBROSIO Introduction In this paper we introduce and study the properties of the class BV(Q, E) of functions of bounded variation u : Q -&#x3E; E, where Q c Rn is an open set and (E, 6) is a locally compact metric space. It is natural to require that for any Lipschitz function p : E -; R and any u E BV(Q, E) the function v = Sp(u) belongs to BV(fl), the classical space of real functions of bounded variation. Moreover, the total variation measure idul has to be greater or equal than IDvl ] provided the Lipschitz constant of p is not greater than 1. We have thus defined E) as the class of Borel functions u : S2 --~ E such that there exists a finite measure u satisfying the condition for any function cp : E -&#x3E; R whose Lipschitz constant is less or equal than 1. The total variation measure ( is the least measure which fulfils (1). It turns out that our definition is consistent with the elementary case Q =]a, b[C R, and IDul agrees with the essential total v

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