Volumeret

Volumeret (the volume measure) is the standard way to assign a nonnegative size to subsets of n-dimensional Euclidean space. In modern analysis, vol_n is typically identified with the Lebesgue measure on R^n, sometimes written as m_n. The vol of a region A is denoted vol_n(A).

Key properties: nonnegativity, vol(empty)=0, monotonicity (A ⊆ B implies vol(A) ≤ vol(B)), translation invariance, and σ-additivity, so that for disjoint measurable sets A_i, vol(⋃ A_i) = Σ vol(A_i). It agrees with intuitive notions of length, area, and ordinary volume in low dimensions; it is complete and outer regular. These features make vol_n a canonical measure for geometric size.

Computing vol: the unit cube in R^n has vol 1; the unit ball B^n has vol π^{n/2} / Γ(n/2+1). For R^3, the sphere volume is 4π/3; for R^2, the disk area is π. In practice, volumes are computed via integration: vol(A) = ∫_{R^n} 1_A(x) dx, and via Fubini's theorem for iterated integrals. For shapes defined by inequalities, vol is the integral of cross-sectional areas.

Relation and limitations: In probability, vol_n(A) is the probability of a uniformly distributed point landing in A. Not all subsets are Lebesgue-measurable; non-measurable sets do not have a defined volume. The concept generalizes to manifolds with volume forms and to other measures, but vol_n serves as the standard reference in Euclidean space.