An original expression for volume in real Euclidean n-space Rⁿ is advanced. The expression depends on a well-ordering produced by a special space-filling-curve in the unit n-cube [0, 1]ⁿ in Rⁿ. The measures induced by the curve, combined with the Heaviside step-function and the Lebesgue integral yield an expression for the volume contained by any closed subset S'(X₁ ,X₂ , X₃, ..., X) = 0 of the unit n-cube [0, 1]ⁿ. This expression for volume in [0, 1]ⁿ is immediately generalized to an expression for volume in Rⁿ by means of a simple transformation which transforms closed bounded subsets S(X₁ , X₂ ,X₃, ... ,Xn) = 0 of Rⁿ onto closed subsets S'(X₁, X₂ ,X₃, ... ,~) = 0 of [0, 1]ⁿ. The generalized expression for volume in Rⁿ is exceedingly simple and involves only a single summation over the positive integers.