Comments on "Constraints on belief functions imposed by fuzzy random variables": some technical remarks on Romer/Kandel.

ABSTRACT

First, we would like to thank V. Kratschmer for his validation of our results in the paper regarding the belief measure by using a topological approach. Though assertions (1) and (3) are presented in a weakened fashion, our results still remain valid, as he claims. It is true that assertion (2) has been proved by us only for Borel sets B, which have at most countable components. We were not able to prove the same result for Borel sets with uncountable components (such as the irrational numbers, for example) using our line of reasoning. We therefore applaud the proof presented by V. Kratschmer for the more general Borel sets using an interesting use of some topological properties induced by the Hansdorff metric defined on the space of closed intervals of the real numbers. This certainly makes our original approach to fuzzy data analysis combining fuzzy sets theory and Dempster-Shafer even more useful.