Plausibility measures on ultrametric spaces
by
Alexander Savchenko
Kherson State Agrarian University

Let P denote a partially ordered set (its partial order is denoted by ≤ ) for which there exist the minimal element ⊥ and the maximal element T ≠ ⊥.

Let B the s-algebra of Borel subsets of a space X. A plausibility measure on X is a map m: B→ P satisfying the following properties: (1) m(∅)=⊥; (2) m(X)=T; (3) A ⊂ B, A, B ∈ B, then m(A) ≤ m(B).

The following is an example of plausibility measure. Let M be a finite nonempty subset of X. We consider the power-set 2^M partially ordered by inclusion. We define the plausibility measure m_M by the formula m_M(A)=M∩A. More generally, every map f: M→ X determines a plausibility measure m_f defined by: m_f(A)=f^-1(A).

We fix P and denote by Pl_f(X) the set of all plausibility measures on X with finite supports on X (we say that the support of m is finite (compact) if there exists a finite (compact) subset A of X such that m(Y)=m(A), for any Y ∈ B).

Let (X, d) be an ultrametric space. For any e > 0, let O_e denote the family of subsets of X that can be represented as unions of (open) balls of radius e. We let

^ d   (m, n)= inf {e > 0 | m(U)=n(U) for any U ∈ Oe}.

Then [^d] is an ultrametric on the set Pl_f(X). The elements of the completion of (Pl_f(X), [^d]) are the plausibility measures with compact supports on X.

We investigate the obtained functor on the category of ultrametric spaces and nonexpanding maps.

Date received: May 1, 2008