Spaces of non-additive measures generated by triangular norms

Kh. Sukhorukova

doi:10.30970/ms.59.2.215-224

Kh. Sukhorukova Ivan Franko National University of Lviv, Lviv, Ukraine

DOI: https://doi.org/10.30970/ms.59.2.215-224

Keywords: non-additive measure; triangular norm; max-min measure; compact Hausdorff space.

Abstract

We consider non-additive measures on the compact Hausdorff spaces, which are generalizations of the idempotent measures and max-min measures. These measures are related to the continuous triangular norms and they are defined as functionals on the spaces of continuous functions from a compact Hausdorff space into the unit segment.
The obtained space of measures (called ∗-measures, where ∗ is a triangular norm) are endowed with the weak* topology. This construction determines a functor in the category of compact Hausdorff spaces. It is proved, in particular, that the ∗-measures of finite support are dense in the spaces of ∗-measures. One of the main results of the paper provides an alternative description of ∗-measures on a compact Hausdorff space X, namely as hyperspaces of certain subsets in X × [0, 1]. This is an analog of a theorem for max-min measures proved by Brydun and Zarichnyi.

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