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 Ph(e) be a mass distribution, i.e. a set function such that: (1) it is defined for Borel sets, (2) it is nonnegative: Ph(e) 0, (3) it is absolutely additive: if e e1 e2 ei ek 0 (i k) then Ph(e) Ph(e1) Ph(e2) Let Ph'(e' be another mass distribution such that Ph(R) Ph'(R) By definition, a translocation of masses is a function Ps(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) Ps(e, R) Ph(e) Ps(R, e' Ph'(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 (Ps, Ph, Ph'