Characterization of the sendograph-convergence of fuzzy sets by means of their Lp- and levelwise convergence

• Authors: W. Trutschnig
• Venue: Fuzzy Sets and Systems
• Year: 2010

One particular way to define a metric on the family of all d-dimensional fuzzy sets with non-empty convex compact @a-cuts is the sendograph metric which measures the Hausdorff distance of the sendographs of fuzzy sets. As first result it will be proved that if a sequence of fuzzy sets converges to a limit fuzzy set with respect to the sendograph metric then it also converges with respect to the L"p-metric and that the converse is true if in addition the supports of the sequence converge to the support of the limit. Additionally, convergence with respect to the sendograph metric will be shown to be equivalent to almost everywhere convergence of the @a-cuts of the sequence to the @a-cut of the limit plus the convergence of the supports of the sequence to the support of the limit (already proved in the one-dimensional setting by Fan). As second and stronger result it will be proved that analogous convergence interrelations also hold for the bigger class of fuzzy sets not necessarily having compact support if the Hausdorff distance is replaced by a metrization of the Fell topology. This in particular implies that the two characterizations proved in the first step do not depend on the choice of the metric used in the definition of the Hausdorff metric. In addition, it is demonstrated that the characterization of weakly compact sets with respect to the sendograph metric as stated by Greco is a straightforward consequence of the abovementioned equivalences and an alternative definition of fuzzy random variables that is equivalent to the common definition is given.