In mathematics, a Caccioppoli set is a subset of ${\displaystyle \mathbb {R} ^{n}}$ whose boundary is (in a suitable sense) measurable and has (at least locally) a finite measure. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli set if its characteristic function is a function of bounded variation, and its perimeter is the total variation of the characteristic function.

The basic concept of a Caccioppoli set was first introduced by the Italian mathematician Renato Caccioppoli in the paper (Caccioppoli 1927): considering a plane set or a surface defined on an open set in the plane, he defined their measure or area as the total variation in the sense of Tonelli of their defining functions, i.e. of their parametric equations, provided this quantity was bounded. The measure of the boundary of a set was defined as a functional, precisely a set function, for the first time: also, being defined on open sets, it can be defined on all Borel sets and its value can be approximated by the values it takes on an increasing net of subsets. Another clearly stated (and demonstrated) property of this functional was its lower semi-continuity.

In the paper (Caccioppoli 1928), he precised by using a triangular mesh as an increasing net approximating the open domain, defining positive and negative variations whose sum is the total variation, i.e. the area functional. His inspiring point of view, as he explicitly admitted, was those of Giuseppe Peano, as expressed by the Peano-Jordan Measure: to associate to every portion of a surface an oriented plane area in a similar way as an approximating chord is associated to a curve. Also, another theme found in this theory was the extension of a functional from a subspace to the whole ambient space: the use of theorems generalizing the Hahn–Banach theorem is frequently encountered in Caccioppoli research. However, the restricted meaning of total variation in the sense of Tonelli added much complication to the formal development of the theory, and the use of a parametric description of the sets restricted its scope.

Lamberto Cesari introduced the "right" generalization of functions of bounded variation to the case of several variables only in 1936: perhaps, this was one of the reasons that induced Caccioppoli to present an improved version of his theory only nearly 24 years later, in the talk (Caccioppoli 1953) at the IV UMI Congress in October 1951, followed by five notes published in the Rendiconti of the Accademia Nazionale dei Lincei. These notes were sharply criticized by Laurence Chisholm Young in the Mathematical Reviews.

In 1952 Ennio De Giorgi presented his first results, developing the ideas of Caccioppoli, on the definition of the measure of boundaries of sets at the Salzburg Congress of the Austrian Mathematical Society: he obtained this results by using a smoothing operator, analogous to a mollifier, constructed from the Gaussian function, independently proving some results of Caccioppoli. Probably he was led to study this theory by his teacher and friend Mauro Picone, who had also been the teacher of Caccioppoli and was likewise his friend. De Giorgi met Caccioppoli in 1953 for the first time: during their meeting, Caccioppoli expressed a profound appreciation of his work, starting their lifelong friendship. The same year he published his first paper on the topic i.e. (De Giorgi 1953): however, this paper and the closely following one did not attract much interest from the mathematical community. It was only with the paper (De Giorgi 1954), reviewed again by Laurence Chisholm Young in the Mathematical Reviews, that his approach to sets of finite perimeter became widely known and appreciated: also, in the review, Young revised his previous criticism on the work of Caccioppoli.

The last paper of De Giorgi on the theory of perimeters was published in 1958: in 1959, after the death of Caccioppoli, he started to call sets of finite perimeter "Caccioppoli sets". Two years later Herbert Federer and Wendell Fleming published their paper (Federer & Fleming 1960), changing the approach to the theory. Basically they introduced two new kind of currents, respectively normal currents and integral currents: in a subsequent series of papers and in his famous treatise, Federer showed that Caccioppoli sets are normal currents of dimension ${\displaystyle n}$ in ${\displaystyle n}$-dimensional euclidean spaces. However, even if the theory of Caccioppoli sets can be studied within the framework of theory of currents, it is customary to study it through the "traditional" approach using functions of bounded variation, as the various sections found in a lot of important monographs in mathematics and mathematical physics testify.

In what follows, the definition and properties of functions of bounded variation in the ${\displaystyle n}$-dimensional setting will be used.

Definition 1. Let ${\displaystyle \Omega }$ be an open subset of ${\displaystyle \mathbb {R} ^{n}}$ and let ${\displaystyle E}$ be a Borel set. The perimeter of ${\displaystyle E}$ in ${\displaystyle \Omega }$ is defined as follows