infinitevolume

In mathematics, infinite volume describes a space or region whose volume, defined by a standard measure such as Lebesgue measure, is infinite. In Euclidean space R^n, the volume of a measurable set E is Vol(E) ∈ [0, ∞], and Vol(E) = ∞ when the measure does not converge. Some familiar infinite-volume regions include the whole space R^n, a half-space, or an infinite cylinder. By contrast, some unbounded regions can have finite volume; for example, the Gabriel’s horn, formed by revolving y = 1/x for x ≥ 1 about the x-axis, has finite volume despite infinite extent.

Volume depends on the chosen measure and geometry. In general metric measure spaces, infinite volume means the measure of the entire space is infinite. Finite versus infinite volume has analytic consequences: integrals over infinite-volume domains may diverge unless the integrand decays sufficiently fast, and probability densities defined on such domains typically require normalization.

In differential geometry, a Riemannian manifold can have finite or infinite volume. Euclidean space has infinite volume, while certain non-compact hyperbolic manifolds can have finite volume due to behavior at infinity; conversely, many non-compact spaces have infinite volume.

In physics, the infinite-volume limit or thermodynamic limit refers to taking the size of a system to infinity to study bulk properties and phase behavior, reducing the influence of boundaries. This limit often necessitates careful limiting procedures, since some quantities converge only in the infinite-volume regime or require renormalization.