On the Translocation of Masses

The original paper was published in Dokl. Akad. Nauk SSSR, 37, No. 7–8, 227–229 (1942). We assume that R is a compact metric space, though some of the definitions and results given below can be formulated for more general spaces. Let Φ(e) be a mass distribution, i.e., a set function such that: (1) it is defined for Borel sets, (2) it is nonnegative: Φ(e) ≥ 0, (3) it is absolutely additive: if e = e1 + e2 + · · · ; ei ∩ ek = 0 (i = k), then Φ(e) = Φ(e1)+ Φ(e2) + · · · . Let Φ′(e′) be another mass distribution such that Φ(R) = Φ′(R). By definition, a translocation of masses is a function Ψ(e, e′) defined for pairs of (B)-sets e, e′ ∈ R such that: (1) it is nonnegative and absolutely additive with respect to each of its arguments, (2) Ψ(e, R) = Φ(e), Ψ(R, e′) = Φ′(e′). Let r(x, y) be a known continuous nonnegative function representing the work required to move a unit mass from x to y. We define the work required for the translocation of two given mass distributions as W (Ψ, Φ, Φ′) = ∫