The structure of fuzzy measure families induced by upper and lower probabilities

On the closure of families of fuzzy measures under eventwise aggregations

“…. , l, for which i∈E (1) C (2) i = Y (2) , where Y (2) = l i=1 C (2) i . This procedure has to be iterated until the initial inequality will become equal to (12) (see Lemma 4), i.e.…”

“…We introduce into consideration the following families of fuzzy measures, which were analyzed in [2,1] (see also [7,8,26,29]): M P is the set of all probability measures on A; 1 We can see that the usual monotonicity coincides with 1-monotonicity. For k = 2 we get the family of fuzzy measures being also called supermodular measures [9] with the characteristic property:…”

“…where P (1) , P (2) ∈ M P , x, y, z ∈ [0, 1] n . The cases, where (y) = 0 or (y) = 1, are considered by analogy.…”

“…The cases, where (y) = 0 or (y) = 1, are considered by analogy. If (y) = 0 then P (1) ∈ M P is chosen arbitrarily; if (y) = 1 then P (2) ∈ M P is chosen arbitrarily. The property is proved in the whole.…”

“…. , l, do not obey the condition from Lemma 4, we can find another set E (1) of indices with the same property as for E and transform (15) to the inequality with a new system of sets C (2) i , i = 1, . .…”