Decomposition Theorem for Finite Perimeter Sets

• Decomposition Theorem for finite perimeter sets provides a canonical partition into indecomposable components that minimizes the total perimeter.
• It employs fine topology, upper gradient relaxation, and flat chain approaches to rigorously establish perimeter additivity in diverse settings.
• The theory extends classical Euclidean results to PI spaces, Wiener spaces, Carnot groups, and nonlocal frameworks, offering robust variational insights.

A set of finite perimeter is a measurable set whose characteristic function has bounded variation, and its boundaries admit a rich structure accessible to analysis, geometry, and topology. The Decomposition Theorem for sets of finite perimeter asserts the existence of a canonical partition into indecomposable components—subsets which cannot be further split without increasing total perimeter—generalizing the classical result in Euclidean spaces to highly general metric and functional analytic settings, including Wiener spaces, PI spaces, Carnot groups, and even nonlocal and abstract settings.

1. Definitions: Finite Perimeter and Indecomposability

Let $(X,d,\mu)$ be a complete metric measure space equipped with a doubling measure and supporting a $1$-Poincaré inequality. A Borel set $E\subset X$ is said to have finite perimeter if its characteristic function $\chi_E$ admits a total variation measure $\|D\chi_E\|$ with $\|D\chi_E\|(X)<\infty$, defined by relaxation via upper gradients or by approximation with Newton–Sobolev functions (Bonicatto et al., 20 Dec 2025). The perimeter $P(E, X)$ is thus

$P(E, X) = \|D\chi_E\|(X).$

A set $E$ is indecomposable if, whenever $E = F \cup G$ (with $F \cap G = \emptyset$, $\mu(F),\mu(G)>0$) into sets of finite perimeter, the perimeter is strictly subadditive, i.e., $P(E, X) < P(F, X) + P(G, X)$. Otherwise, if equality holds, $E$ is decomposable. In Euclidean spaces, indecomposability corresponds to measure-theoretic connectedness; more generally, it is characterized topologically by connectivity in the $1$-fine topology (Bonicatto et al., 20 Dec 2025).

2. Decomposition Theorems in Metric and Analytic Settings

The decomposition theorem states: Let $E\subset X$ be a set of finite perimeter. Then there exists a countable (or finite) partition into Borel sets $\{E_i\}_{i\in I}$ such that:

• Each $E_i$ is indecomposable;
• The sets $E_i$ are pairwise disjoint modulo $\mu$-null sets;
• $E = \bigcup_{i=1}^\infty E_i$ up to a null set;
• The perimeter is additive:

$P(E, X) = \sum_{i=1}^\infty P(E_i, X)$

• The partition is unique up to null sets.

This result holds in PI spaces (doubling, Poincaré), with sharpness—that is, dropping the Poincaré assumption can lead to failure of the decomposition (Lahti, 2021, Bonicatto et al., 2019). The removal of earlier isotropicity hypotheses (i.e., perimeter density being independent of the subset) demonstrates the robustness of the variational approach via upper gradients (Lahti, 2021).

3. Proof Strategies and Key Techniques

Two principal approaches establish the decomposition theorem:

A. Fine Topology Approach

The essential connected components of $E$ can be identified via the $1$-fine topology—the finest topology compatible with the variational 1-capacity. The measure-theoretic interior $I_E$ is decomposed into its $1$-finely connected components, each indecomposable. The two-sidedness property ensures the disjointness of boundaries and exact perimeter additivity (Bonicatto et al., 20 Dec 2025).

B. Relaxation and Upper Gradients (Variational Approach)

Directly working with upper gradients, optimal Lipschitz approximations, and localization arguments yields additivity propositions, enabling Zorn-type maximality arguments to produce the indecomposable splitting without recourse to Hausdorff measure representations or isotropicity (Lahti, 2021).

C. Abstract Decomposition via Flat Chains

In the context of rectifiable normal G-flat chains (more general objects than sets), an abstract decomposition principle is proved using monotone convergence and a superlinear norm inequality, dispensing with compactness. The result yields a setwise splitting into set-indecomposable subchains, which specializes to the decomposition of finite-perimeter sets (Goldman et al., 2022).

4. Extensions: Nonlocal, Group, and Wiener Space Decompositions

Nonlocal Perimeter Functionals:

For Gagliardo-type and distributional Caccioppoli nonlocal perimeters, decomposition into components is determined by an $\varepsilon$-connectivity condition—partitioning into maximal $\varepsilon$-connected subsets, each $\varepsilon$-indecomposable. As $\varepsilon\to 0$, the decomposition and variational structure $\Gamma$-converges to the classical local setting (Carioni et al., 7 Feb 2025).

Wiener/Gaussian Spaces:

In separable Banach spaces equipped with a Gaussian measure (“Wiener spaces”), the structure theorem for sets of finite perimeter establishes that, at almost every perimeter point, the blow-up of $E$ converges in measure to a halfspace. The reduced boundary is rectifiable by Gaussian–Lipschitz hypersurfaces (Ambrosio et al., 2012).

Group-Theoretic Settings (Heisenberg, Carnot):

In Carnot groups and the Heisenberg group (discrete or continuous), the decomposition theorem applies via Ahlfors regularity and group structure, with intrinsic Lipschitz graphs and corona decompositions representing the local geometry of perimeter sets (Naor et al., 2017).

5. Connections to Extreme Points and Convex Structure in BV

Indecomposable sets correspond to extreme points in the unit ball of BV seminorms. Choquet-type characterizations show that the normalized indicators of indecomposable (sometimes “simple”) sets are precisely the extreme points of the BV ball, with ramifications for imaging and variational analysis (Bonicatto et al., 2019, Carioni et al., 7 Feb 2025).

6. Practical Implications, Sharpness, and Examples

The decomposition theorem is sharp: the PI (doubling, Poincaré) structure is necessary, as counterexamples arise in doubling spaces lacking Poincaré inequalities (Lahti, 2021). Special cases include Euclidean spaces (recovering Ambrosio–Caselles–Masnou–Morel), Carnot groups, RCD(K,N) spaces, Wiener spaces for Gaussian analysis, and nonlocal/capacitary settings. Fine topological insights confirm that indecomposability coincides precisely with fine connectedness (Bonicatto et al., 20 Dec 2025).

7. Tabular Summary: Main Settings and Methods

Setting	Main Methodology	Key Reference
PI metric measure spaces	Upper gradient relaxation	(Lahti, 2021)
Wiener (Gaussian) spaces	Blow-up to half-spaces	(Ambrosio et al., 2012)
Abstract G-flat chains	Abstract decomposition lemma	(Goldman et al., 2022)
Nonlocal perimeters	$\varepsilon$-connected splitting	(Carioni et al., 7 Feb 2025)
Heisenberg/Carnot groups	Intrinsic Lipschitz, corona	(Naor et al., 2017, Bonicatto et al., 2019)
Fine topology (general metric)	1-fine connected components	(Bonicatto et al., 20 Dec 2025)

The decomposition theorem for sets of finite perimeter constitutes a foundational result in geometric measure theory, functional analysis, and metric geometry, enabling detailed structure theorems and fine analysis of variational problems. Its extensions to nonlocal, infinite-dimensional, group-theoretic and abstract settings reflect its universality in the analytic and geometric study of boundaries.