Indecomposable sets of finite perimeter in doubling metric measure spaces

We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak (1,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1,1)$$\end{document}-Poincaré inequality. The two main results we obtain are a decomposition theorem into indecomposable sets and a characterisation of extreme points in the space of BV functions. In both cases, the proof we propose requires an additional assumption on the space, which is called isotropicity and concerns the Hausdorff-type representation of the perimeter measure.


Introduction
The classical Euclidean theory of functions of bounded variation and sets of finite perimeter-whose cornerstones are represented, for instance, by [6,15,17,22,29,36] -has been successfully generalised in different directions, to several classes of metric structures. Amongst the many important contributions in this regard, we just single out the pioneering works [9][10][11]16,25,28]. Although the basic theory of BV functions can be developed on abstract metric measure spaces (see, e.g., [5]), it is in the framework of doubling spaces supporting a weak (1, 1)-Poincaré inequality (in the sense of Heinonen-Koskela [30]) that quite a few fine properties are satisfied (see [1,2,38]).
The aim of the present paper is to study the notion of indecomposable set of finite perimeter on doubling spaces supporting a weak (1, 1)-Poincaré inequality (that we call PI spaces for brevity). By indecomposable set we mean a set of finite perimeter E that cannot be written as disjoint union of two non-negligible sets F, G satisfying P(E) = P(F) + P(G). This concept constitutes the measure-theoretic counterpart to the topological notion of 'connected set' and, as such, many statements concerning connectedness have a correspondence in the context of indecomposable sets.
In the Euclidean framework, the main properties of indecomposable sets have been systematically investigated by Ambrosio et al. in [4]. The results of this paper are mostly inspired by (and actually extend) the contents of [4]. In the remaining part of the Introduction, we will briefly describe our two main results: the decomposition theorem for sets of finite perimeter and the characterisation of extreme points in the space of BV functions. In both cases, the natural setting to work in is that of PI spaces satisfying an additional condition-called isotropicity-which we are going to describe in the following paragraph.
Let (X, d, m) be a PI space and E ⊂ X a set of finite perimeter; we refer to Sect. 1 for the precise definition of perimeter and the terminology used in the following. The perimeter measure P(E, ·) associated to E can be written as θ E H ∂ e E , where H stands for the codimension-one Hausdorff measure on X, while ∂ e E is the essential boundary of E (i.e., the set of points where neither the density of E nor that of its complement vanishes) and θ E : ∂ e E → [0, +∞) is a suitable density function; cf. Theorem 1.23. The integral representation formula was initially proven in [1] only for Ahlfors-regular spaces, and this additional assumption has been subsequently removed in [2]. It is worth to point out that the weight function θ E might (and, in some cases, does) depend on the set E itself; see, for instance, Example 1.27. In this paper, we mainly focus our attention on those PI spaces where θ E is independent of E, which are said to be isotropic (the terminology comes from [7]). As we will discuss in Example 1.31, the class of isotropic PI spaces includes weighted Euclidean spaces, Carnot groups of step 2 and non-collapsed RC D spaces. Another key feature of the theory of sets of finite perimeter in PI spaces is given by the relative isoperimetric inequality (see Theorem 1.17 below), which has been obtained by M. Miranda in the paper [38].
Our main result (namely, Theorem 2.14) states that on isotropic PI spaces any set of finite perimeter E can be written as (finite or countable) disjoint union of indecomposable sets. Moreover, these components-called essential connected components of E-are uniquely determined and maximal with respect to inclusion, meaning that any indecomposable subset of E must be contained (up to null sets) in one of them. We propose two different proofs of this decomposition result, in Sects. 2 and 4, respectively. The former is a variational argument that was originally carried out in [4], while the latter is adapted from [33] and based on Lyapunov's convexity theorem. However, both approaches strongly rely upon three fundamental ingredients: representation formula for the perimeter measure, relative isoperimetric inequality, and isotropicity. We do not know whether the last one is in fact needed for the decomposition to hold (see also Example 2.16).
Furthermore, in Sect. 3 we study the extreme points in the space BV(X) of functions of bounded variation defined over X; we are again assuming (X, d, m) to be an isotropic PI space. More precisely: call K(X; K ) the family of all those functions f ∈ BV(X) supported in K , whose total variation satisfies |D f |(X) ≤ 1 (where K ⊂ X is a fixed compact set). Then we can completely characterise (under a few additional assumptions) the extreme points of K(X; K ) as a convex, compact subset of L 1 (m); see Theorem 3.8. It turns out that these extreme points coincide (up to a sign) with the normalised characteristic functions of simple sets (cf. Definition 3.1). In the Euclidean case, the very same result was proven by W. H. Fleming in [23,24] (see also [13]). Part of Sect. 3 is dedicated to some equivalent definitions of simple set: in the general framework of isotropic PI spaces, a plethora of phenomena concerning simple sets may occur, differently from what happens in R n (see [4]). For more details, we refer to the discussion at the beginning of Sect. 3.1. Given an open set ⊂ X and any E ⊂ , we write E to specify that E is bounded and dist(E, X\ ) > 0.

Functions of bounded variation
In the framework of general metric measure spaces, the definition of function of bounded variation-which is typically abbreviated to 'BV function'-has been originally introduced in [38] and is based upon a relaxation procedure. Let us recall it: (1.1) Then f is said to be of bounded variation-briefly, f ∈ BV(X)-if f ∈ L 1 (m) and |D f |(X) < +∞.
We can extend the function |D f | defined in (1.1) to all Borel sets via Carathéodory construction: This way we obtain a finite Borel measure |D f | on X, which is called the total variation measure of f .
loc (m) be a sequence satisfying sup n |D f n |(X) < +∞. Then there exist a subsequence (n i ) i and some f ∞ ∈ L 1 loc (m) such that f n i → f ∞ in L 1 loc (m).
It follows from item (i) of Proposition 1.2 that the space BV(X) is a Borel subset of L 1 (m).

Remark 1.3
Let (X, d, m) be a metric measure space. Fix f ∈ BV(X) and m > 0. Then We thus conclude that which yields the statement.
We conclude this subsection by briefly recalling an alternative (but equivalent) approach to the theory of BV functions on abstract metric measure spaces, which has been proposed in [18,19].
A derivation over a metric measure space (X, d, m) is a linear map b : LIP bs (X) → L 0 (m) such that the following properties are satisfied: (ii) Weak locality. There exists a non-negative function G ∈ L 0 (m) such that The least function G (in the m-a.e. sense) having this property is denoted by |b|.
The space of all derivations over (X, d, m) is denoted by Der(X). The support spt(b) ⊂ X of a derivation b ∈ Der(X) is defined as the essential closure of the set |b| = 0 . Given any b ∈ Der(X) with |b| ∈ L 1 loc (m), we say that div(b) ∈ L p for some p ∈ [1, ∞] provided there exists a (necessarily unique) function div For a proof of the above representation formula, we refer to [18,Theorem 7.3.4].

Sets of finite perimeter
The study of sets of finite perimeter on abstract metric measure spaces has been initiated in [38] (where, differently from here, the term 'Caccioppoli set' is used). In this subsection we report the definition of set of finite perimeter and its basic properties, more precisely the ones that are satisfied on any metric measure space (without any further assumption).
In particular, it holds that P(E, ·) = P(F, ·) whenever m(E F) = 0. (ii) Lower semicontinuity. The function P(·, ) is lower semicontinuous with respect to the L 1 loc (m )-topology: namely, if (E n ) n is a sequence of Borel subsets of such that the convergence 1 E n → 1 E holds in L 1 loc (m ) as n → ∞, then P(E, ) ≤ lim n P(E n , ).
(v) Compactness. Let (E n ) n be a sequence of Borel subsets of X with sup n P(E n ) < +∞.
Then there exist a subsequence (n i ) i and a Borel set E ∞ ⊂ X such that 1 E n i → 1 E ∞ in the L 1 loc (m)-topology as i → ∞.

Fine properties of sets of finite perimeter in PI spaces
The first aim of this subsection is to recall the definition of PI space and its main properties; we refer to [31] for a thorough account about this topic. Thereafter, we shall recall the definition of essential boundary and the main properties of sets of finite perimeter in PI spaces-among others, the isoperimetric inequality, the coarea formula, and the Hausdorff representation of the perimeter measure. Finally, we will discuss the class of isotropic PI spaces, which plays a central role in the rest of the paper.
for every x ∈ X and r > 0.
The least such constant C D is called the doubling constant of (X, d, m).

Remark 1.9
Let (X, d, m) be a doubling metric measure space. Then spt(m) = X. Indeed, it holds that m B r (x) > 0 for every x ∈ X and r > 0, otherwise m would be the null measure. Moreover, the metric space (X, d) is proper (i.e., bounded closed subsets of X are compact).
Doubling spaces do not have a definite dimension (not even locally), but still are 'finitedimensional'-in a suitable sense. In light of this, it makes sense to consider the codimensionone Hausdorff measure H, defined below via Carathéodory construction, which takes into account the local change of dimension of the underlying space. Definition 1.10 (Codimension-one Hausdorff measure) Let (X, d, m) be a doubling metric measure space. Given any set E ⊂ X and any parameter δ > 0, we define g dm for every x ∈ X and r > 0, for every x ∈ X and r > 0, (1.4) where the constants C P and λ are chosen as in Definition 1.12.
Proof A standard diagonalisation argument provides us with a sequence ( Given that the local Lipschitz constant lip( f n ) is an upper gradient of the function f n , it holds that Moreover, for some function g ∈ L 1 (m B r (x) ) we have (up to a not relabelled subsequence) that f n (y) ≤ g(y) for every n ∈ N and m-a.e. y ∈ B r (x). We can further assume that f n (y) → f (y) for m-a.e. y ∈ B r (x). Given that f n (y) − ( f n ) x,r ≤ g(y) + g x,r for every n ∈ N and m-a.e. y ∈ B r (x), we deduce (by dominated convergence theorem) that Therefore, by letting n → ∞ in (1.5) we conclude that the claim (1.4) is verified.
For the purposes of this paper, we shall only consider the following notion of PI space (which is strictly more restrictive than the usual one, where a weak (1, p)-Poincaré inequality is required for some exponent p that is possibly greater than 1): Definition 1.14 (PI space) We say that a metric measure space (X, d, m) is a PI space provided it is doubling and satisfies a weak (1, 1)-Poincaré inequality.
We introduce the concept of essential boundary in a doubling metric measure space and its main features. The discussion is basically taken from [1,2], apart from a few notational discrepancies.
Given a doubling metric measure space (X, d, m), a Borel set E ⊂ X and a point x ∈ X, we define the upper density of E at x and the lower density of E at x as respectively. Whenever upper and lower densities coincide, their common value is called density of E at x and denoted by D(E, x). We define the essential boundary of the set E as It clearly holds that the essential boundary ∂ e E is contained in the topological boundary ∂ E. Moreover, we define the set E 1 /2 ⊂ ∂ e E of points of density 1/2 as Finally, we define the essential interior E 1 of E as

Remark 1.15
Let F ⊂ E ⊂ X be given. Then Hence, the claim (1.6) is proven.
The following result is well-known. We report here its full proof for the reader's convenience.
Proof (i) It trivially stems from the very definition of essential boundary.
In order to prove that even the inclusion ∂ e (E ∩ F) ⊂ ∂ e E ∪ ∂ e F is verified, it is just sufficient to combine the previous case with item (i): Hence, the proof of (1.7) is complete.
In the setting of PI spaces, functions of bounded variation and sets of finite perimeters present several fine properties, as we are going to describe.
for every x ∈ X and r > 0, where s > 1 is any exponent greater than log 2 (C D ).
As an immediate consequence of Theorem 1.17, we have that a given set of finite perimeter E in a PI space (X, d, m) has null perimeter if and only if either m(E) = 0 or m(E c ) = 0. Proof As proven in [38], there exists a constant C I > 0 such that for every x ∈ X and r > 0. By letting r → +∞ in (1.10), we conclude that (1.9) is satisfied. A function f ∈ BV(X) is said to be simple provided it can be written as f = n i=1 λ i 1 E i , for some λ 1 , . . . , λ n ∈ R and some sets of finite perimeter E 1 , . . . , E n ⊂ X having finite m-measure. It holds that any function of bounded variation in a PI space can be approximated by a sequence of simple BV functions (with a uniformly bounded total variation), as we are going to state in the next well-known result. Nevertheless, we recall the proof of this fact for the sake of completeness.
Proof Given that f m : for all m > 0 by Remark 1.3, it suffices to prove the statement under the additional assumption that the function f is essentially bounded, say that −k < f < k holds m-a.e. for some k ∈ N. Let us fix any n ∈ N. Given any i = −kn + 1, . . . , kn, we can choose Then we define the simple BV function f n on X as It can be readily checked that |D f n |(X) ≤ |D f |(X). Indeed, notice that Therefore, it holds that thus accordingly | f − f n | = | f − i/n| ≤ 1/n on E i,n for all i = −kn, . . . , kn. This ensures that Since spt( f n ) ⊂ K for every n ∈ N by construction, the proof of the statement is achieved.

Remark 1.22
In the proof of Lemma 1.21 we obtained a stronger property: each approximating function f n (say, The following result states that, in the context of PI spaces, the perimeter measure admits an integral representation (with respect to the codimension-one Hausdorff measure): where τ ∈ (0, 1/2) is a constant depending just on C D , C P and λ. Moreover, the set (1.14) We shall sometimes consider θ E as a Borel function defined on the whole space X, by declaring that θ E := 0 on the set X\∂ e E. Proof By using item (iii) of Proposition 1.7 we deduce that

Proof
The density function θ E that appears in the Hausdorff representation formula for P(E, ·) might depend on the set E itself (cf. Example 1.27 below for an instance of this phenomenon). On the other hand, the new results that we are going to present in this paper require the density θ E to be 'universal'-in a suitable sense. The precise formulation of this property is given in the next definition, which has been proposed in [7, Definition 6.1].

Definition 1.25 (Isotropic space)
Let (X, d, m) be a PI space. Then we say that (X, d, m) is isotropic provided for any pair of sets E, F ⊂ X of finite perimeter satisfying F ⊂ E it holds that In order to provide examples and counterexamples, it will be convenient to consider the metric measure space we are going to construct. Given any n ∈ N, we define the n-spider S n as (1. 16) We say that o is the origin of S n , while R 1 , . . . , R n are the rays of S n . We identify o with (i, 0) for every i = 1, . . . , n. It holds that (S n , d, m) is an Ahlfors-regular PI space, where d is given by and m stands for the 1-dimensional Hausdorff measure on (S n , d (1.17) Proof Possibly replacing E with its complement E c -an operation which does not affect P(E, ·) nor ∂ e E-we can suppose without loss of generality that k ≤ n − k, thus λ = k. The first statement readily follows from the fact that P(E, ·) is absolutely continuous with respect to the counting measure on S n . Indeed, the Ahlfors-regularity of (S n , d, m) grants that H is equivalent to the 0-dimensional Hausdorff measure, whence Theorem 1.23 yields the previous claim. To prove the last statement, first observe that-since each ray R i can be identified with (0, +∞)-it holds It thus remains to characterise P(E, ·) for some open neighbourhood of o. To this aim, take any r ∈ (0, 1/2) such that {i} × (0, r ) is m-a.e. contained in E for every i ∈ I and call := B r (o). It is then clear by construction that ∂ e E ∩ ⊂ {o}. Therefore, in order to prove (1.17) it just suffices to show that P(E, ) = k. On the one hand, we define the sequence On the other hand, fix any sequence ( f j ) j ⊆ LIP loc ( ) satisfying f j → 1 E in L 1 loc (m ) and for which the limit lim j lip( f j ) dm exists. Up to a subsequence, we can also assume that f j → 1 E in the m -a.e. sense. Now let ε ∈ (0, r ) be given. Then for any j ∈ N sufficiently large we can find points By first letting j → ∞ and then ε → 0, we finally conclude that lim j lip( f j ) dm ≥ k and thus accordingly P(E, ) = k. Therefore, the proof of the last statement is completed.
We shall also sometimes work with PI spaces satisfying the following property: The same arguments show that, given a radius r > 0, the closure X r of the ball B r (o) in S 3 (endowed with the restricted distance d X r ×X r and measure m X r ) is an isotropic, Ahlforsregular PI space which does not have the two-sidedness property.
A sufficient condition for the two-sidedness property to hold is provided by the following result: be a PI space with the following property: This proves the two-sidedness property. Example 1.31 (Examples of isotropic spaces) Let us conclude the section by expounding which classes of PI spaces are known to be isotropic (to the best of our knowledge): (i) Weighted Euclidean spaces (induced by a continuous, strong A ∞ weight). (ii) Carnot groups. (iii) RC D(K , N ) spaces, with K ∈ R and N < ∞. In particular, all (weighted) Riemannian manifolds whose Ricci curvature is bounded from below.
Isotropicity of the spaces in (i) and (ii) is shown in [7, Section 7] and [8], respectively. Also, it follows from the rectifiability results in [26,27] that all Carnot groups of step 2 satisfy (1.19), so also the two-sidedness property. About item (iii), it follows from the results in [3,14] that all RC D(K , N ) spaces satisfy (1.19), whence they have the two-sidedness property (and thus are isotropic).

Decomposability of a set of finite perimeter
This section is entirely devoted to the decomposability properties of sets of finite perimeter in isotropic PI spaces. An indecomposable set is, roughly speaking, a set of finite perimeter that is connected in a measure-theoretical sense. Section 2.1 consists of a detailed study of the basic properties of this class of sets. In Sect. 2.2 we will prove that any set of finite perimeter can be uniquely expressed as disjoint union of indecomposable sets (cf. Theorem 2.14). The whole discussion is strongly inspired by the results of [4], where the decomposability of sets of finite perimeter in the Euclidean setting has been systematically investigated. Actually, many of the results (and the relative proofs) in this section are basically just a reformulation-in the metric setting -of the corresponding ones in R n , proven in [4]. We postpone to Remark 2.19 the discussion of the main differences between the case of isotropic PI spaces and the Euclidean one.

Definition of decomposable set and its basic properties
Let us begin with the definition of decomposable set and indecomposable set in a general metric measure space.

Definition 2.1 (Decomposable and indecomposable sets)
Observe that the property of being decomposable/indecomposable is invariant under modifications on m-null sets and that any m-negligible set is indecomposable.

Remark 2.2
Let E ⊂ X be a set of finite perimeter. Let {E n } n∈N be a partition of E into sets of finite perimeter and let ⊂ X be any open set. Then it holds that: Indeed, it can be readily checked that 1 n≤N E n → 1 E in L 1 loc (m) as N → ∞, whence items (ii) and (iii) of Proposition 1.7 grant that the inequality is always verified.

Lemma 2.3
Let (X, d, m) be an isotropic PI space. Let E, F ⊂ X be sets of finite perimeter and let B ⊂ X be any Borel set. Then the following implications hold: .
Given that θ E∪F is assumed to be null on the complement of ∂ e (E ∪ F), we deduce that Accordingly, it holds that and, similarly, that P(F, Hence, we conclude that Consequently, we have finally proven that H(∂ e E ∩∂ e F ∩ B) = 0, as required.
(ii) Let us suppose that m(E ∩ F) = 0 and H(∂ e E ∩ ∂ e F ∩ B) = 0. We already know that the inequality P(E ∪ F, B) ≤ P(E, B) + P (F, B) is always verified. The converse inequality readily follows from our assumptions, item (iv) of Proposition 1.16 and the representation formula for the perimeter measure: Therefore, it holds that P(E ∪ F, B) = P(E, B) + P (F, B), as required.
In the setting of PI spaces having the two-sidedness property, the fact of being an indecomposable set of finite perimeter can be equivalently characterised as illustrated by the following result, which constitutes a generalisation of [21, Proposition 2.12]. (X, d, m) be a PI space. Then the following properties hold: Suppose (X, d, m) has the two-sidedness property. Then any indecomposable subset of X satisfies (2.1).

Theorem 2.5 Let
Proof (i) Suppose E ⊂ X is decomposable. Choose two disjoint sets of finite perimeter F, G ⊂ X having positive m-measure such that E = F ∪ G and P(E) = P(F) + P(G). Then let us consider the function f := 1 F ∈ L 1 loc (m). Notice that |D f |(X) = P(F) < +∞. Moreover, we know from Lemma 1.24 that H(∂ e F\∂ e E) = 0, thus accordingly Nevertheless, f is not m-a.e. equal to a constant on E, whence E does not satisfy property (2.1). (ii) Fix an indecomposable set E ⊂ X. Consider any function f ∈ L 1 loc (m) such that |D f |(X) < +∞ and |D f |(E 1 ) = 0. First of all, we claim that Indeed, by exploiting the inclusion ∂ e (E ∩ A) ⊂ ∂ e E ∪ ∂ e A and the isotropicity of (X, d, m) we get holds for a.e. t ∈ R. Therefore, item (ii) of Lemma 2.3 grants that P(E) = P(E + t ) + P(E\E + t ) for a.e. t ∈ R. Being E indecomposable, we deduce that for a.e. t ∈ R we have that either m( Let us define t − , t + ∈ R as follows: Being open, it holds 1 ⊃ , whence |D f |( ) = 0. Given any x ∈ , we can choose a radius r > 0 such that B λr (x) ⊂ and accordingly |D f | B λr (x) = 0, where λ ≥ 1 is the constant appearing in the weak (1, 1)-Poincaré inequality. Consequently, Lemma 1.13 tells us that constant on B r (x). This shows that f is locally m-a.e. constant on . Since is connected, we deduce that f is m-a.e. constant on . Therefore, we finally conclude that is indecomposable by using item (i) of Theorem 2.5.
Now fix any set A ⊂ E of finite perimeter. By using again the property (1.7) we see that On the other hand, we claim that This shows the validity of (2.6).

P(F ∩ E n , ) for every Borel set F ⊂ E with P(F) < +∞.
Proof Fix any N ∈ N. By repeatedly applying Lemma 2.8 we obtain that By letting N → ∞ we deduce that P(F, ) ≥ ∞ n=0 P(F ∩E n , ), which gives the statement thanks to Remark 2.2.

Proposition 2.10 (Stability of indecomposable sets) Let (X, d, m) be an isotropic PI space. Fix a set E ⊂ X be of finite perimeter. Let (E n ) n be an increasing sequence of indecomposable subsets of X such that E = n E n . Then E is an indecomposable set.
Proof We argue by contradiction: suppose there exists a Borel partition {F, G} of the set E such that m(F), m(G) > 0 and P(E) = P(F) + P(G). Given that we have lim n m(F ∩ E n ) = m(F) and lim n m(G ∩ E n ) = m(G), we can choose an index n ∈ N so that m(F ∩ E n ), m(G ∩ E n ) > 0. By Lemma 2.8 we know that P(E n ) = P(F ∩ E n ) + P(G ∩ E n ). Being {F ∩ E n , G ∩ E n } a Borel partition of E n , we get a contradiction with the indecomposability of E n . This gives the statement.

Proof
whence P(E, B) = P(F, B)+P(G, B) by item (ii) of the same lemma. This is in contradiction with the fact that E is indecomposable in B, thus the statement is proven.

Decomposition theorem
The aim of this subsection is to show that any set of finite perimeter in an isotropic PI space can be uniquely decomposed into indecomposable sets.

Remark 2.12
Let {a n i } i,n∈N ⊂ (0, +∞) be a sequence that satisfies lim n a n i = a i for every i ∈ N and lim j lim n ∞ i= j a n i = 0. Then ∞ i=0 a i = lim n ∞ i=0 a n i . Indeed, for every j ∈ N we have that whence by letting j → ∞ we conclude that ∞ i=0 a i ≤ lim n ∞ i=0 a n i ≤ lim n ∞ i=0 a n i ≤ ∞ i=0 a i , which proves the claim.

Let us denote by P the collection of all Borel partitions
Note that the family P is non-empty, as it contains the element (E ∩ , ∅, ∅, . . .).

Let us call
(2.9) Choose any (E n i ) i∈N n ⊂ P such that lim n ∞ i=0 μ(E n i ) = M. Since P(E n i ) ≤ P(E) + P( ) < +∞ for every i, n ∈ N, we know by the compactness properties of sets of finite perimeter that we can extract a (not relabelled) subsequence in n in such a way that the following property holds: there exists a sequence (E i ) i∈N of Borel subsets of E ∩ such (2.10) Given any i, j ∈ N such that i = j, we also have that μ(E i ∩ E j ) = lim n μ(E n i ∩ E n j ) = 0, thus accordingly m(E i ∩ E j ) = 0. Moreover, by lower semicontinuity of the perimeter we see that and, similarly, that ∞ i=0 P(E i ) ≤ P(E) + P( ) for every i ∈ N. To prove that (E i ) i∈N ∈ P it only remains to show that m (E ∩ )\ i E i = 0. We claim that Observe that the inequality m(E n i ) ≤ m( \E n i ) holds for every i ≥ 1. Let us define We readily deduce from the relative isoperimetric inequality (1.8) that for all j ≥ 1 we have Furthermore, by using the previous estimate and again (1.14) we obtain that Consequently, we deduce that the claim (2.11) is verified. By recalling also (2.10) and Remark 2.12, we can conclude that This forces m (E ∩ )\ i E i = 0 and accordingly (E i ) i∈N ∈ P. Hence, in other words (E i ) i∈N is a maximiser for the problem in (2.9). Finally, we claim that each set E i is indecomposable in . Suppose this was not the case: then for some j ∈ N we would find a partition {F, G} of E j into sets of finite perimeter having positive m-measure and satisfying the identity P(E j , ) = P(F, ) + P(G, ). We can relabel the family {E i } i = j ∪ {F, G} as (F i ) i∈N in such a way that m(F i ) i∈N is a non-increasing sequence. Given that we see that (F i ) i∈N ∈ P. On the other hand, given that α > 1 and μ(F), μ(G) > 0 we have the inequality μ(F) This leads to a contradiction with (2.9), whence the sets E i are proven to be indecomposable in . Therefore, the family {E i } i∈I , where I := i ∈ N : m(E i ) > 0 , satisfies the required properties.
Maximality. Let F ⊂ E ∩ be a fixed Borel set with P(F) < +∞ that is indecomposable in . Choose an index j ∈ I for which m(F ∩ E j ) > 0. By Corollary 2.9 we know that Given that F is assumed to be indecomposable in , we finally conclude that F ∩ i = j E i has null m-measure, so that m(F\E j ) = 0. This shows that the elements of {E i } i∈I are maximal.
Uniqueness. Consider any other family {F j } j∈J having the same properties as {E i } i∈I . By maximality we know that for any i ∈ N there exists a (unique) j ∈ N such that m(E i F j ) = 0, thus the two partitions {E i } i∈I and {F j } j∈J are essentially equivalent (up to m-negligible sets). This proves the desired uniqueness.
We are now ready to prove the main result of this section: Proof Letx ∈ X be a fixed point. Choose a sequence of radii r j +∞ such that j := B r j (x) has finite perimeter for all j ∈ N. Let us apply Proposition 2.13: given any j ∈ N, there exists an m-essentially unique partition {E j i } i∈I j of E ∩ j , into sets of finite perimeter that are maximal indecomposable subsets of j , with m(E j i ) > 0 for all i ∈ I j and P(E, j ) = i∈I j P(E j i , j ). Given any j ∈ N and i ∈ I j , we know from Lemma 2.11 that E j i is indecomposable in j+1 , thus there exists ∈ I j+1 for which m(E (2.13) Given any x ∈ E, let us define the set G x ⊂ E One clearly has that m(G x ) > 0. Moreover, it readily follows from (2.13) that for every j 0 ∈ N. (2.14) We claim that: For every x, y ∈ E it holds that either G x ∩ G y = ∅ or G x = G y . (2.15) In order to prove it, assume that G x ∩ G y = ∅ and pick any z ∈ G x ∩ G y . Then there exist some indices j x , j y ∈ N, i x ∈ I j x and i y ∈ I j y such that {x, z} ⊂ E This shows that the sets {E i } i∈I have finite perimeter, while the fact that they are indecomposable follows from Proposition 2.10. Now fix any finite subset J of I . Similarly to the estimates above, we see that for every j ≥ max j(i) : i ∈ J it holds that . By arbitrariness of J ⊂ I this yields the inequality i∈I P(E i ) ≤ P(E), thus accordingly P(E) = i∈I P(E i ) by Remark 2.2. Finally, maximality and uniqueness can be proven by arguing exactly as in Proposition 2.13. Therefore, the statement is achieved.

Example 2.16
Although we do not know if the Decomposition Theorem 2.14 holds without the assumption on isotropicity, one can see that the assumption on (1, 1)-Poincaré inequality cannot be relaxed to a (1, p)-Poincaré inequality with p > 1. As an example of this, one can take a fat Sierpiński carpet S a ⊂ [0, 1] 2 with a sequence a ∈ 2 \ 1 , as defined in [35]. The set S a , equipped with a natural measure m and distance d, is a 2-Ahlfors-regular metric measure space supporting a (1, p)-Poincaré inequality for all exponents p > 1. Nevertheless, given any vertical strip of the form I x,ε :

Remark 2.19
Let us highlight the two main technical differences between the proofs we carried out in this section and the corresponding ones for R n that were originally presented in [4]: (i) There exist isotropic PI spaces X where it is possible to find a set of finite perimeter E whose associated perimeter measure P(E, ·) is not concentrated on E 1 /2 . For instance, consider the space described in Example 1.29: it is an isotropic PI space where (1.18) fails, thus in particular property (1.19) is not verified (as a consequence of Lemma 1.30). Some of the results of [4]-which have a counterpart in this paper-are proven by using property (1.19). Consequently, the approaches we followed to prove some of the results of this section provide new proofs even in the Euclidean setting. (ii) An essential ingredient in the proof of the decomposition theorem [4, Theorem 1] is the (global) isoperimetric inequality. In our case, we only have the relative isoperimetric inequality at disposal, thus we need to 'localise' the problem: first we prove a local version of the decomposition theorem (namely, Proposition 2.13), then we obtain the full decomposition by means of a 'patching argument' (as described in the proof of Theorem 2.14). Let us point out that in the Ahlfors-regular case the proof of the decomposition theorem would closely follows along the lines of [4, Theorem 1] (thanks to Theorem 1.18).
Finally, an alternative proof of the decomposition theorem will be provided in Sect. 4.

Extreme points in the space of BV functions
The aim of Sect. 3.1 is to study the extreme points of the 'unit ball' in the space of BV functions over an isotropic PI space (with a uniform bound on the support). More precisely, given an isotropic PI space (X, d, m) and a compact set K ⊂ X, we will detect the extreme points of the convex set made of all functions f ∈ BV(X) such that spt( f ) ⊂ K and |D f |(X) ≤ 1, with respect to the strong topology of L 1 (m); cf. Theorem 3.8. Informally speaking, the extreme points coincide-at least under some further assumptions-with the (suitably normalised) characteristic functions of simple sets, whose definition is given in Definition 3.1. In Sect. 3.2 we provide an alternative characterisation of simple sets (cf. Theorem 3.17) in the framework of Alhfors-regular spaces, a key role being played by the concept of saturation of a set, whose definition relies upon the decomposition properties treated in Sect. 2.

Simple sets and extreme points in BV
A set of finite perimeter E ⊂ R n having finite Lebesgue measure is a simple set provided one of the following (equivalent) properties is satisfied: (i) E is indecomposable and saturated, the latter term meaning that the complement of E does not have essential connected components of finite Lebesgue measure. (ii) Both E and R n \E are indecomposable. (iii) If F ⊂ R n is a set of finite perimeter such that ∂ e F is essentially contained in ∂ e E (with respect to the (n − 1)-dimensional Hausdorff measure) and L n (F) < +∞, then it holds that F = E (up to L n -null sets).
We refer to [4,Section 5] for a discussion about the equivalence of the above conditions. In the more general setting of isotropic PI spaces, (the appropriate reformulations of) these three notions are no longer equivalent. The one that well captures the property we are interested in (i.e., the fact of providing an alternative characterisation of the extreme points in BV) is item (iii), which accordingly is the one that we choose as the definition of simple set in our context: It is rather easy to prove that-under some additional assumptions-the definition of simple set we have just proposed is equivalent to (the suitable rephrasing of) item (ii) above: Proof (i) Assume E ⊂ X is a simple set. First, we prove by contradiction that E is indecomposable: suppose it is not, thus it can be written as E = F ∪ G for some pairwise disjoint sets F, G of finite perimeter such that m(F), m(G) > 0 and P(E) = P(F)+P(G). By combining item (iv) of Proposition 1.16 with item (ii) of Lemma 2.3, we obtain that In particular, we have that H(∂ e F\∂ e E) = H(∂ e G\∂ e E) = 0. Being E simple, we get F = G = E, which leads to a contradiction. Then E is indecomposable. In order to show that also E c is indecomposable, we argue in a similar way: suppose E c = F ∪G for pairwise disjoint sets F , G of finite perimeter with m(F ), m(G ) > 0 and P(E c ) = P(F )+P(G ). By arguing as before we obtain that H(∂ e E c \∂ e F ) = H(∂ e E c \∂ e F ) = 0. Being ∂ e E c = ∂ e E, we can conclude (again since E is simple) that F = G = E c , whence the contradiction. Therefore, E c is indecomposable.
(ii) Assume that (X, d, m) has the two-sidedness property and that E, E c are indecomposable sets. Take a set F ⊂ X of finite perimeter such that H(∂ e F\∂ e E) = 0. We know from (1.18) that Consequently, we deduce that

Remark 3.3
In item (ii) of Proposition 3.2, the additional assumption on the space cannot be dropped. To show it, let us consider the closed unit ball X 1 centered at o of the 3-spider (S 3 , d, m). We claim that the conclusion of item (ii) of Proposition 3.2 fails in (X 1 , d X 1 ×X 1 , m X 1 ).
By Example 1.29 we know that the two-sidedness property is not satisfied. Now call R 1 , R 2 , R 3 the intersections of the rays of S 3 with X 1 . It thus holds that R 1 ∪ R 2 and X 1 \(R 1 ∪ R 2 ) = R 3 ∪ {o} are indecomposable, but the set R 1 ∪ R 2 is not simple.
Let (X, d, m) be a metric measure space. Let K ⊂ X be a compact set. Then we define

Remark 3.4 It holds that
First of all, its convexity is granted by item (ii) of Proposition 1.2. To prove compactness, fix any sequence ( f n ) n ⊂ K(X; K ). Item (iii) of Proposition 1.2 says that f n i → f in L 1 loc (X) for some subsequence (n i ) i and some limit function f ∈ L 1 loc (m). Given that spt( f n ) ⊂ K for every n ∈ N, we know that spt( f ) ⊂ K , thus f ∈ L 1 (m) and f n i → f in L 1 (m). Finally, by using item (i) of Proposition 1.2 we conclude that |D f |(X) ≤ lim i |D f n i |(X) ≤ 1, whence f ∈ K(X; K ).
Recall that ext K(X; K ) stands for the set of all extreme points of K(X; K ); cf. "Appendix A". Furthermore, observe that |D f |(X) = 1 holds for every f ∈ ext K(X; K ). In the remaining part of this subsection, we shall study in detail the family ext K(X; K ). Our arguments are strongly inspired by the ideas of the papers [23,24].
Let (X, d, m) be a PI space. Given a set E ⊂ X of finite perimeter satisfying 0 < m(E) < +∞ and m(E c ) > 0 (so that P(E) > 0), let us define ∈ BV(X).
Observe that D + (E) (X) = D − (E) (X) = 1. For any compact set K X, we define Observe that S(X; K ), I(X; K ) ⊂ F (X; K ) ⊂ K(X; K ). Given any function f ∈ F (X; K ), we shall denote by E f ⊂ X the (m-a.e. unique) Borel set satisfying either Proposition 3.5 Let (X, d, m) be a PI space and K X a compact set. Then the closed convex hull of the set F (X; K ) coincides with K(X; K ).
Proof We aim to show that any function f ∈ K(X; K ) can be approximated in L 1 (m) by convex combinations of elements in F (X; K ). Let us apply Lemma 1.21: we can find a sequence ( f n ) n of simple BV functions supported on the set K , say f n = k n i=1 λ n i 1 E n i , so that f n → f in L 1 (m) and k n i=1 |λ n i | P(E n i ) ≤ 1 (recall Remark 1.22). Given that we have sgn(λ n i ) (E n i ) ∈ F (X; K ) and we conclude that the functions f n /q belong to the convex hull of F (X; K ). Given that F (X; K ) is symmetric, we know that its convex hull contains the function 0 and accordingly also all the functions f n . The statement follows.

Lemma 3.6 Let (X, d, m) be a PI space and let K X be a compact set. Then it holds that
converging to some function f ∈ L 1 (m) in L 1 (m). We aim to show that f ∈ B as well. Given any n ∈ N, we can find λ n ∈ [−1, 1] and a set of finite perimeter E n ⊂ K such that m(E n ) > 0 and f n = λ n + (E n ). We subdivide the proof into three different cases: (1) Suppose lim n P(E n ) = 0. Then there exists a set of finite perimeter E ⊂ K such that (up to a not relabelled subsequence) it holds 1 E n → 1 E in L 1 (m). In particular, P(E) ≤ lim n P(E n ) = 0 and accordingly lim n m(E n ) = m(E) = 0. Possibly passing to a further subsequence, we may thus assume that m(E n ) < 1/2 n for all n ∈ N. Since the identity f k (x) = 0 holds for every k ≥ n > 0 and x ∈ K \ m≥n E m , we deduce that lim k f k (x) = 0 for every x ∈ K \ n m≥n E m . This implies that f = 0 ∈ B, as the set n m≥n E m is m-negligible by Borel-Cantelli lemma.
(2) Suppose that lim n P(E n ) = +∞. Then it holds that (3) Suppose lim n P(E n ) > 0 and lim n P(E n ) < +∞. Then there exist λ ∈ [−1, 1] and c ∈ (0, +∞) such that-up to a not relabelled subsequence-one has that λ n → λ and lim n P(E n ) = c. We can further assume that 1 E n → 1 E strongly in L 1 (m), for some set of finite perimeter E ⊂ K . Therefore, we deduce that f = λ1 E /c. If m(E) = 0, then f = 0 ∈ B. If m(E) > 0, then we can write f as λ + (E), where we set λ := λ P(E)/c. Since P(E) ≤ lim n P(E n ) = c by lower semicontinuity of the perimeter, we conclude that λ ∈ [−1, 1] and accordingly f ∈ B.
Theorem 3.7 Let (X, d, m) be a PI space and let K X be a compact set. Then it holds that ext K(X; K ) ⊂ I(X; K ).
Proof By Milman Theorem A.1, Proposition 3.5, and Lemma 3.6, we know that any extreme point of K(X; K ) can be written as λ f for some λ ∈ [−1, 1] and f ∈ F (X; K ). Moreover, it is clear that It only remains to prove that if f = σ (E) ∈ ext K(X; K ), then E is indecomposable. We argue by contradiction: suppose the set E is decomposable, so that there exist disjoint Borel sets F, G ⊂ E such that m(F), m(G) > 0 and P(E) = P(F) + P(G). Therefore, we can write This contradicts the fact that f is an extreme point of K(X; K ), thus the set E is proven to be indecomposable. Hence, we have that ext K(X; K ) ⊂ I(X; K ), as required. This implies that B\A = B − A is Borel measurable and thus B\A ∈ D. (c) Given any increasing sequence (A n ) n ⊂ D, we have that |Dg| n A n = lim n |Dg|(A n ) for all g ∈ L 1 (m) thanks to the continuity from below, whence accordingly ∪ n A n is Borel measurable (so that n A n ∈ D) as it is the pointwise limit of A n as n → ∞.
All in all, we have proven that D is a Dynkin system. Given that the topology of (X, d) is contained in D (again by item (i) of Proposition 1.2), we conclude that D coincides with the Borel σ -algebra of X by the Dynkin π-λ Theorem. This proves that B is Borel for any B ⊂ X Borel, as claimed.
Given that |Dg|(X) = 1 for every g ∈ ext K(X; K ), we know that |Dg|(X) = 1 for μ-a.e. g ∈ L 1 (m) and accordingly ν is a probability measure. Now fix any open set ⊂ X containing ∂ e E f . Thanks to Theorem 1.4, we can find a sequence of derivations (b n ) n ⊂ Der b (X) such that |b n | ≤ 1 in the m-a.e. sense, spt(b n ) and Given that |D f | and ν are outer regular, we can pick a sequence ( n ) n of open subsets of X containing ∂ e E f such that |D f |(∂ e E f ) = lim n |D f |( n ) and ν(∂ e E f ) = lim n ν( n ). By recalling the inequality (3.3), we thus obtain that This forces the equality |Dg|(∂ e E f ) dμ(g) = ν(∂ e E f ) = 1. Given that |Dg|(∂ e E f ) ≤ 1 holds for μ-a.e. g ∈ L 1 (m), we infer that actually |Dg|(∂ e E f ) = 1 for μ-a.e. g ∈ L 1 (m).
Since E f is a simple set, we deduce that m(E g E f ) = 0 for μ-a.e. g ∈ L 1 (m). This forces 1]. Given that μ is concentrated on the symmetric set ext K(X; K ), we finally conclude that f ∈ ext K(X; K ), as required. This proves the inclusion (3.1).
(ii) Let f ∈ ext K(X; K ) be fixed. Take a set F ⊂ X of finite perimeter with H(∂ e F\∂ e E f ) = 0. We claim that either m(F\K ) = 0 or m(F c \K ) = 0. To prove it, notice that (1.7), (1.18) give

Remark 3.11
Given an isotropic PI space (X, d, m) such that m(X) = +∞, it holds that any simple set E ⊂ X is indecomposable and saturated. Indeed, item (i) of Proposition 3.2 grants that both E, E c are indecomposable; since m(E c ) = +∞, we also conclude that E has no holes.   (2.16) = 0, thus proving the statement.
Let us now focus on the special case of Ahlfors-regular, isotropic PI spaces. In this context, simple sets can be equivalently characterised as those sets that are both indecomposable and saturated (cf. Theorem 3.17). In order to prove it, we need some preliminary results: Proposition 3.14 Let (X, d, m) be a k-Ahlfors regular, isotropic PI space with k > 1. Suppose that m(X) = +∞. Let E ⊂ X be an indecomposable set such that m(E) < +∞. Then there exists exactly one essential connected component F ∈ CC e (E c ) satisfying m(F) = +∞.
Proof Let us prove that at least one essential connected component of E c has infinite mmeasure. We argue by contradiction: suppose m(E i ) < +∞ for all i ∈ I , where we set CC e (E) = {E i } i∈I . In particular, we have that m(E c i ) = +∞ holds for every i ∈ I , whence Theorem 1.18 yields By using the Markov inequality we deduce that J := i ∈ I : m(E i ) ≥ 1 is a finite family, thus the set i∈J E i has finite m-measure. This leads to a contradiction, as it implies that Hence, there exists i ∈ I such that m(E i ) = +∞. Suppose by contradiction to have m(E j ) = +∞ for some j ∈ I \{i}. Then E j ⊂ E c i and accordingly m(E c i ) = +∞, which is not possible as we have that min m(E i ), m(E c i ) ≤ C I P(E i ) k /k−1 < +∞ by Theorem 1.18. The statement follows.

Remark 3.15
The Ahlfors-regularity assumption in Proposition 3.14 cannot be dropped, as shown by the following example. Let us consider the strip X := R × [0, 1] ⊂ R 2 , endowed with the (restricted) Euclidean distance and the 2-dimensional Hausdorff measure, which is an isotropic PI space. Then the square E := [0, 1] 2 ⊂ X is an indecomposable set having finite measure, but its complement consists of two essential connected components having infinite measure. (X, d, m) is a k-Ahlfors regular, isotropic PI space with k > 1 and m(X) = +∞, then for any indecomposable set E ⊂ X with m(E) < +∞ it holds that m sat(E) < +∞.

Remark 3.16 If
Indeed, we know that m sat(E) c = +∞ by Proposition 3.14, whence the set sat(E) must have finite m-measure (otherwise we would contradict Theorem 1.18). Proof Necessity stems from Remark 3.11. To prove sufficiency, suppose that E is indecomposable and saturated. Proposition 3.14 grants that E c is the unique element of CC e (E) having infinite m-measure, thus in particular E c is indecomposable. By applying item (ii) of Proposition 3.2, we finally conclude that the set E is simple, as desired.
for simplicity, cf. Remark 4.6 for a few comments about the unbounded case). The inspiration for this approach is taken from [33]. We refer to "Appendix B" for the language and the results we are going to use in this section.
Let (X, d, m) be an isotropic PI space. Given any open set ⊂ X and any set E ⊂ having finite perimeter in X, we define the family (E) as Observe that The proof of this fact is a direct consequence of the very definition of indecomposable set.  This forces the equality P(E, ) = P(F ∪ G, ) + P E\(F ∪ G), . Given that F ∪ G has finite perimeter, we have proved that F ∪ G ∈ (E). This shows that (E) is closed under finite unions. Finally, to prove that (E) is closed under countable unions, fix any (F i ) i ⊂ (E). Calling F := i∈N F i , we aim to prove that F ∈ (E). We denote F i := F 1 ∪ · · · ∪ F i ∈ (E) for all i ∈ N. Given that F = i∈N F i , we have 1 F i → 1 F and 1 E\F i → 1 E\F in L 1 loc (m ). Hence, by lower semicontinuity and subadditivity of the perimeter we can conclude that This says that the set F has finite perimeter in X, thus F belongs to (E), as desired.
To prove the last statement, let us assume that E . By exploiting Remark 1.20 and the boundedness of E, we can find an open ball B ⊂ X such that E B and H(∂ B) < +∞, thus accordingly the family (E) = B (E) is a σ -algebra by the previous part of the proof. (X, d, m) be an isotropic PI space and ⊂ X an open set. Then we claim that
Therefore, the statement is achieved. Proof We can assume without loss of generality that = B r (x) for somex ∈ X and r > 0. For the sake of brevity, let us denote M F := (F, (F), m F ) for every F ∈ (E). It follows from Remark 4.3 that (E) F = (F) and that the atoms of M F coincide with the atoms of M E that are contained in F. Accordingly, in order to prove that M E is purely atomic, it suffices to show that M F is atomic for any set F ∈ (E) with m(F) > 0. We argue by contradiction: suppose M F is non-atomic. Let us fix any ε > 0. Corollary B.5 grants that there exists a finite partition {F 1 , . . . , By letting ε 0 in (4.1) we get that P(F, ) = +∞, which yields a contradiction. Therefore, we conclude that the measure space M F is non-atomic, as required.
Proof First of all, let us prove the following claim: Given any set A ∈ A with μ(A) > 0 and any ε > 0, there exists B ∈ A such that B ⊂ A and 0 < μ(B) < ε. (B.1) In order to prove it, fix a subset A ∈ A of A with 0 < μ(A ) < +∞ (whose existence follows from the semifiniteness assumption) and any k ∈ N such that k > μ(A )/ε. Since μ admits no atoms, we can find a partition B 1 , . . . , B k ∈ A of A such that μ(B i ) > 0 for every i = 1, . . . , k. Hence, there must exist i = 1, . . . , k such that μ(B i ) < ε, otherwise we would have that μ(A ) = μ(B 1 ) + · · · + μ(B k ) ≥ k ε > μ(A ).

Remark B.4
Given a semifinite, non-atomic measure space (X, A, μ) and a set E ∈ A, it holds that (E, A E , μ E ) is semifinite and non-atomic as well, where the restricted σalgebra A E is defined as A E := A ∩ E : A ∈ A . In particular, one can readily deduce from Theorem B.3 that for any λ ∈ 0, μ(E) there exists A ∈ A E such that μ(A) = λ. (X, A, μ) be a finite, non-atomic measure space. Then for every ε > 0 there exists a partition {A 1 , . . . , A n } ⊂ A of X such that μ(A i ) ≤ min ε, μ(A c i ) for all i = 1, . . . , n.