Lebesguemitta

Lebesguemitta is a fictional construct in measure theory used in speculative discussions about extending the Lebesgue measure. It envisions a family of measures on subsets of Euclidean space that blends the standard Lebesgue size with a boundary-regularization term called mitta, intended to reflect how a set’s boundary contributes to its overall size.

For a bounded Borel set A ⊂ R^n, Lebesguemitta μ(A) is defined by μ(A) = Leb(A) + τ · M(A), where Leb is the Lebesgue measure, τ is a fixed real parameter, and M(A) is a boundary complexity term defined by M(A) = lim_{ε→0} m(∂A_ε)/ε, if the limit exists; ∂A_ε denotes the ε-neighborhood of the boundary ∂A, and m denotes Lebesgue measure. Intuitively, μ assigns the usual volume to sets with smooth boundaries but adds a boundary penalty for irregular boundaries.

Properties include translation-invariance (for fixed τ) and monotonicity with respect to set inclusion. If the boundary is smooth and ∂A has measure zero, M(A) = 0 and μ(A) = Leb(A). For fractal or highly irregular boundaries, M(A) can be positive, leading to μ(A) > Leb(A).

Variants consider a one-parameter family μ_r with μ_r(A) = Leb(A) + τ_r M(A), or extensions to probability measures on other spaces using the same boundary term.

History and usage: the term arose in online mathematics folklore as a teaching thought experiment to explore how adding a boundary term would interact with standard measure properties and measurability questions. It is not part of formal measure theory and has no official references.

See also: Lebesgue measure, boundary, outer measure, inner measure.