Non-additive Measures, Set Distances and Cost Functions on Sets: A Frechet-Nikodym-Aronszajn Distance and Cost Function

Torra V.

4th International Conference on Intelligent and Fuzzy Systems (INFUS), Bornova, Turkey, 19 - 21 July 2022, vol.504, pp.7-15 identifier identifier

  • Publication Type: Conference Paper / Full Text
  • Volume: 504
  • Doi Number: 10.1007/978-3-031-09173-5_2
  • City: Bornova
  • Country: Turkey
  • Page Numbers: pp.7-15
  • Keywords: Set functions, Non-additive measures, Fuzzy measues, Cost functions
  • Yıldız Technical University Affiliated: No


Fuzzy measures are set functions that are monotonic with respect to the set inclusion. They are also called non-additive measures, monotonic games, and capacities. Cost functions, as the ones used in the optimal transport problem, are typically functions between pairs of elements of, in general, two different sets. In some particular problems, they can be functions that take two elements from the same set. In this case, it is usual that the cost from one element to itself is zero and between different elements is not zero. Other properties can vary, but it is not uncommon that the cost function satisfies the axioms of a distance when applied to two elements of the same set.