Monotone measures-based integrals

The theory of classical measures and integral reflects the genuine property of several quantities in standard physics and/or geometry, namely the σ-additivity. Though monotone measures not assuming σ-additivity appeared naturally in models extending the classical ones (for example, inner and outer measures, where the related integral was considered by Vitali already in 1925), their intensive research was initiated in the past 40 years by the computer science applications in areas reflecting human decisions, such as economy, psychology, multicriteria decision support, etc. In this chapter, we summarize basic types of monotone measures together with the basic monotone measures-based integrals, including the Choquet and Sugeno integrals, and we introduce the concept of universal integrals proposed by Klement et al. to give a common roof for all mentioned integrals. Benvenuti’s integrals linked to semicopulas are shown to be a special class of universal integrals. Up to several other integrals, we also introduce decomposition integrals due to Even and Lehrer, and show which decomposition integrals are inside the framework of universal integrals.