The fuzzy set-valued measures generated by fuzzy random variables

Some Characterizations of the Choquet Integral with Respect to a Monotone Interval-Valued Set Function

“…means almost everywhere in the Lebesgue measure m, and |g| is the absolute value of g [15,16]. In [13,23], we can see that (A) F dm is a nonempty bounded and closed interval in R whenever F is an interval-valued function as in the following theorem.…”

“…In the past decade, it has been suggested to use intervals in order to represent uncertainty, for example, for economic uncertainty [12], for fuzzy random variables [13], in intervalprobability [14], for martingales of multi-valued functions [15], in the integrals of set-valued functions [16], in the Choquet integrals of interval-valued (or closed set-valued) functions [17][18][19][20][21][22], and for interval-valued capacity functions [23]. Couso-Montes-Gil [24] studied applications under the sufficient and necessary conditions on monotone set functions, i.e., the subadditivity of the Choquet integral with respect to monotone set functions.…”

On Fuzzy Measure on Fuzzy sets

“…The notion of fuzzy measure was extended by Avallone and Barbieri, Jiang and Suzuki [14] Narukawa and Murofushi [8] , Ralescu and Adams [10] as a set function which was defined on σ-field with values in [0, ] . After that, many authors studied the fuzzy measure and proved some results about it as Guo and Zhang [8] Kui [13], Li and Yasuda [6] Lushu and Zhaohu [15] Minghu [16].…”