DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Canonical TV level sets and their boundaries

Definition

Assume Countable Choice. Let $u \in L_{l o c}^{1} (R^{2}; R)$ , with functions identified when they agree almost everywhere. For full-space TV, use $u$ itself. For a bounded-domain Dirichlet or Neumann problem, use the zero extension $\tilde{u}$ only to define the geometric level sets; this does not change the selected regularizer, and for each nonzero level the resulting set lies inside $Ω$ . For $t > 0$ , the positive $t$ -level class is the measurable-set class of $E_{t}^{+} (u) : = {x \in R^{2} : u (x) > t}$ ; for $t < 0$ , the negative $t$ -level class is the class of $E_{t}^{-} (u) : = {x \in R^{2} : u (x) < t}$ . These classes do not depend on the representative of $u$ . For any measurable set $E \subseteq R^{2}$ , its density-one representative is

$E^{(1)} : = {x \in R^{2} : \lim_{r ↓ 0} \frac{∣ E \cap B (x, r) ∣}{∣ B (x, r) ∣} = 1} .$

This representative depends only on the almost-everywhere class of $E$ . The canonical level set $U_{u}^{(t)}$ is $(E_{t}^{+} (u))^{(1)}$ when $t > 0$ and $(E_{t}^{-} (u))^{(1)}$ when $t < 0$ . Its canonical boundary is the ordinary topological boundary $\partial U_{u}^{(t)}$ in $R^{2}$ . For two nonempty boundary sets, use the usual extended Hausdorff distance induced by Euclidean distance; set $d_{H} (\emptyset, \emptyset) = 0$ and $d_{H} (E, F) = + \infty$ if exactly one set is empty. Hausdorff convergence uses this convention and is invariant under changes to a function on a null set.

Definition references

Spatial BV and finite perimeter are defined on almost-everywhere classes.

Zero extension and the distinct Dirichlet and Neumann total-variation conventions.

The Hausdorff distance uses the Euclidean metric on $R^{2}$ .

The Lebesgue density theorem supplies density-one representatives.

Iglesias, Mercier, and Scherzer, A note on convergence of solutions of total variation regularized linear inverse problems, §3

Chambolle, Duval, Peyré, and Poon, Geometric properties of solutions to the total variation denoising problem, §2.1

Depends on

Used by

Dependency tree · two levels

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Sources

José A. Iglesias, Gwenael Mercier, and Otmar Scherzer, A note on convergence of solutions of total variation regularized linear inverse problems, §3 (standard reference, not scraped)
Antonin Chambolle, Vincent Duval, Gabriel Peyré, and Clarice Poon, Geometric properties of solutions to the total variation denoising problem, §2.1 (standard reference, not scraped)