In the theory of fuzzy measures, Choquet integral is one of the most important tools. The calculation of Choquet integral on real line is difficult for many cases such as nonmonotone functions. In this paper, by the geometric interpretation of Choquet integral, we introduce a new Choquet-like integral with a different algebraic interpretation of Choquet integral on real line. The calculation of this integral is simpler than Choquet integral on real line. Based on this integral, we introduce a general class of normal distribution on monotone measures. Finally, as an application, the real dataset obtained from the daily price of Dow Jones Industrial Average Index in period of June 2, 2008 to June 2, 2018 is analyzed.

In this paper, a numerical Monte Carlo integration for Choquet integrals is proposed by using a generalized version of mean value theorem based on Choquet integral. In special cases, this generalization can help us to have the classical Monte Carlo integration and the mean value theorem over some unbounded regions.

In probability theory, mathematical expectation of a random variable is very important. Choquet expectation (integral), as a generalization of mathematical expectation, is a powerful tool in various areas, mainly in generalized probability theory and decision theory. In vector spaces, combining Choquet expectation and Pettis integral has led to a challenging and an interesting subject for researchers. In this paper, we indicate and discuss a failure in the previous definition of Choquet-Pettis integral of Banach space-valued functions. To obtain a correct definition of Choquet-Pettis integral, an open problem concerning the linearity of the Choquet integral is stated.

The Choquet integral and the Sugeno integral provide a useful tool in many problems in engineering and social choice where the aggregation of data is required. In this paper, previous results of Hong (Nonlinear Analysis 2011 74:7296–7303) are improved by relaxing some of their requirements. Carlson's, Sandor's, Bushell–Okrasinski's type inequalities and Fatou's lemma for universal integral are studied in a rather general form, thus generalizing some recent results. © 2012 Wiley Periodicals, Inc.

The aim of the present paper is to establish an extended Chebyshev type inequality for semi(co)normed fuzzy integrals based on an aggregation function and a scale transformation. The extended Chebyshev type inequality given in this paper provides a generalization of some previous results. Finally, some conclusions are drawn and some problems for further investigations are given.