volimit

Volimit is a term used in geometry and analysis to denote the limiting value of volumes under a sequence of operations or transformations. It may refer to the limit of Lebesgue measures of a sequence of sets, or more generally to the limit of a family of volume-like quantities in a metric-measure framework. The exact definition can vary by context, and in some discussions volimit is treated as a general notion rather than a fixed, universally standardized concept.

In its simplest form, if (X, μ) is a measure space and {A_n} is a sequence of measurable subsets with finite measure, the volimit of {A_n} is defined as lim μ(A_n) when the limit exists. If the limit does not exist, one can speak of the lower volimit as lim inf μ(A_n) and the upper volimit as lim sup μ(A_n). When the sequence is monotone, standard measure continuity theorems yield the volimit automatically.

In settings with varying metrics d_n or different ambient spaces, volimit can refer to the limiting value of μ_n(X) under a suitable notion of convergence, such as measured Gromov-Hausdorff convergence. Variants of volimit appear in geometric measure theory, analysis on metric spaces, and models of material porosity, where one studies the stability of volume under deformations or coarse approximations.

Examples help illustrate the idea: letting A_n = [0,1/n] in the real line with Lebesgue measure gives a volimit of 0. Let A_n be unions of intervals whose total length approaches a constant L; the volimit would be L in that scenario.

See also: limit of measures, volume, measure theory, Gromov-Hausdorff convergence. Notes: the term volimit is not universally standardized; different authors may define or emphasize different aspects.