ON THE TRANSLOCATION OF MASSES

摘要

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 = e_1 + e_2 + ···; e_i ∩ e_k = 0 (i ≠ k), then Φ(e) = Φ(e_1) + Φ(e_2) + ···. 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) = Ψ(e), Ψ(R, e') = Φ'(e').