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Fuzzy measures are monotonic set functions on a reference set; they generalize probabilities replacing the additivity condition by monotonicity. The typical application of these measures is with fuzzy integrals. Fuzzy integrals integrate a function with respect to a fuzzy measure, and they can be used to aggregate information from a set of sources (opinions from experts or criteria in a multicriteria decision-making problem). In this context, background knowledge on the sources is represented by means of the fuzzy measures. For example, interactions between criteria are represented by means of nonadditive measures. In this paper, we introduce fuzzy measures on multisets. We propose a general definition, and we then introduce a family of fuzzy measures for multisets which we show to be equivalent to distorted probabilities when the multisets are restricted to proper sets.