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On exact and totally bounded capacities

Taras Radul

Abstract

We consider capacity (fuzzy measure, non-additive probability) on a compactum as a monotone cooperative normed game. We introduce topological analogues of well known classes of exact and totally balanced games and show that these classes form subfunctors of the capacity functor which lie between known subfunctors of convex capacities and balanced capacities. It is natural to consider probability measures as elements of core of such games. We describes exact capacities as a retraction of the convex closed sets of probability measures. Using such representation we prove openness of the functor of exact capacities.

On exact and totally bounded capacities

Abstract

We consider capacity (fuzzy measure, non-additive probability) on a compactum as a monotone cooperative normed game. We introduce topological analogues of well known classes of exact and totally balanced games and show that these classes form subfunctors of the capacity functor which lie between known subfunctors of convex capacities and balanced capacities. It is natural to consider probability measures as elements of core of such games. We describes exact capacities as a retraction of the convex closed sets of probability measures. Using such representation we prove openness of the functor of exact capacities.
Paper Structure (4 sections, 10 theorems, 5 equations)

This paper contains 4 sections, 10 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Exact and totally balanced capacities
  3. Categorical properties of balanced capacities
  4. Openness of the functor of exact capacities

Key Result

Lemma 1

The set $MTBX$ is closed and convex in $MX$.

Theorems & Definitions (13)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Lemma 2
  • Definition 3
  • Lemma 3
  • Theorem 1
  • Theorem 2
  • Lemma 4
  • Proposition 1
  • ...and 3 more