Set of finite perimeter ($\mathrm{Per}(E;\Omega )=|D{\chi }_E|(\Omega )$)

A measurable set $E$ has finite perimeter in an open domain $\Omega$ when the distributional derivative of ${\chi }_E$ is a finite vector Radon measure there:The indicator is always locally integrable. Equivalently it belongs locally to BV space and has finite total derivative variation on the whole domain. If $|E\cap \Omega |<\infty$, it also belongs to the global BV space with its L1 norm. Finite perimeter itself does not require that finite-volume condition.

For example, the half-line $(0,\infty )$ has perimeter one in $\mathbb{R}$, from its single endpoint Dirac measure, despite infinite length. The entire real line has perimeter zero, although its indicator is not in $L^1(\mathbb{R})$. On a bounded domain, finite volume is automatic, so the simpler condition ${\chi }_E\in BV(\Omega )$ is equivalent.

Relative perimeter counts the interface inside $\Omega$ and excludes boundary outside that open domain. In a Caccioppoli partition, each shared interior interface appears in the perimeters of both adjacent regions; a segmentation cost must account for that double counting.

The perimeter of a measurable set relative to an open domain is the total variation seminorm of its indicator function there. For a smooth region it equals the Hausdorff measure of the boundary lying inside the domain. Global perimeter uses the entire space as the domain; an internal interface shared by two regions appears in both relative perimeters and must be counted only once in a segmentation energy.

Here the sets of finite perimeter $E,F$ differ only compactly inside $U$. A minimizing set with bounded volume forcing satisfies this inequality. Interior regularity theorems give graph patches outside a singular set negligible for surface Hausdorff measure; bounded forcing allows the graph estimates used in the noncontact of ROF level boundaries.

The total variation seminorm of the indicator function within an open set counts only its interior interface. The perimeter of a zero extension may add a boundary contribution.

The BV lattice inequality $|D\min (v,w)|+|D\max (v,w)|\le |Dv|+|Dw|$ applied to two indicator functions gives the perimeter inequality. For smooth approximations, minima and maxima partition the two gradients; sequential lower semicontinuity passes the estimate to BV. It prevents a crossing of minimizers with strictly ordered set forcing.

Suppose $E_g$ minimizes $\mathrm{Per}(E)-{\int }_{E}g$ and $E_h$ minimizes $\mathrm{Per}(E)-{\int }_{E}h$. Compare them with their intersection and union. Adding their optimality inequalities and using submodularity of relative perimeter gives ${\int }_{E_g\setminus E_h}(h-g)\le 0$. Strict ordering forces that set to be null. Applied to $g=(f-t)/\alpha$, this yields nested ROF denoising superlevel sets.

A Caccioppoli partition is a countable measurable partition, up to null sets, into sets of finite perimeter with finite sum of relative perimeters. Its interior interfaces are counted twice by that sum. It represents regions of a piecewise constant image signal in the piecewise-constant Mumford–Shah problem.