In order to face with the complexity of discrete fuzzy measures, we propose the concept of k-
orderadditive fuzzy measure, including usual additive measures and fuzzy measures. Every
discrete fuzzy measure is a k-order additive fuzzy measure for a unique k. A related topic of
the paper is to introduce an alternative representation of fuzzy measures, called the
interaction representation, which sets and extends in a common framework the Shapley
value and the interaction index proposed by Murofushi and Soneda.