Multivalued Functions Integration: from Additive to Arbitrary Non-negative Set Function Endre Pap Abstract It is given a short overview of some integrals of multifunctions based on additive measures, as strong, Aumann and Aumann-Gould integrals. It is con- sidered also a multi-valued Choquet integral based on a multisubmeasure. Then it is introduced a set-valued Gould type integral of multifunctions with values in the family of all nonempty bounded subsets of a real Banach space X and with respect to an arbitrary non-negative set function. There are given some basic properties of the integrable multifunctions, and some continuity properties of the multimeasure induced by set-valued integral. 1 Introduction Theory of multifunctions, i.e., set-valued maps, correspondences, etc., is important field of investigations as theoretical and practical applications [1, 2, 4, 5, 9–11, 20, 21, 32, 42, 55, 60]. It allows one to take into account the multiplicity of possible choices, the lack of information and/or the uncertainty in a lot of situations ranging from Optimal Control to Economic Theory, see [3, 31, 60]. In particular, measur- able multifunctions, i.e., set-valued random variables, random sets, are investigated in probability and statistics, with many applications, see first papers [38, 54]. Var- ious types of integrals for multifunctions have many applications in mathematical economics, theory of control, probabilities. Integrals of multifunctions can be used as an aggregation tool when dealing with a large amount of information fusing and with data mining problems such as programming and classification. In processes of subjective evaluation, for instance, the integral of a multifunction can be a tool in synthetic evaluation of the quality of a given object, when the score function may E. Pap (B) Singidunum University, Danijelova 32, 11000 Belgrade, Serbia e-mail: epap@singidunum.ac.rs E. Pap Óbuda University, Becsi út 92, Budapest 1034, Hungary © Springer International Publishing Switzerland 2016 S. Saminger-Platz and R. Mesiar (eds.), On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory, Studies in Fuzziness and Soft Computing 336, DOI 10.1007/978-3-319-28808-6_15 257