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\title{On universal integrals}
\author{Jana Molnárová} 
\address{Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University, Jesenná 5, 041 54 Košice, Slovakia}
\email{jana.molnarova@upjs.sk}
\label{molnarova}
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The Choquet and the Sugeno integral provide a suitable tool for solving many problems connected with aggregation of data in engineering and social sciences. The construction of the Choquet integral is based on standard arithmetic operations of addition and multiplication, therefore each Choquet integral-based aggregation is co-monotone additive. On the other hand, the~Sugeno integral is based on the lattice operations of join (maximum) and meet (minimum), thus the corresponding aggregation is join- or meet-homogeneous. Because of the used operations in construction of both the integrals, their applicability is restricted. This fact motivated the paper~\cite{Mesiar}, where a construction of universal integral generalizing both the Choquet and the Sugeno case is presented. See also~\cite{Boczek}. This new concept of~integral is universal in the sense that it allows to integrate non-negative real-valued functions with respect to arbitrary (non-negative) monotone set functions in such a way that covers both the Choquet~(\ref{ch}), the Sugeno integral~(\ref{su}) and also the Shilkret integral~(\ref{sh}) which can be regarded as an ancestor of the Sugeno integral.
\begin{align}
\label{ch}\textbf{Ch}(m,f) & = \int_0^\infty m(\{f\geq t\})\,\mathrm{d}t\\
\label{su}\textbf{Su}(m,f) & = \sup_{t\in[0,\infty]}\{\min(t,m(\{f\geq t\}))\}\\
\label{sh}\textbf{Sh}(m,f) & = \sup_{t\in[0,\infty]}\{t\cdot
m(\{f\geq t\})\}\
\end{align}

Universal integrals are related to general operations of pseudo-multi\-pli\-cation and pseudo-addition. A different approach to construction of universal integrals is based on copulas and can be perceived as an expression of dependence of~values of integrable functions and values of measure with respect to which we integrate. Certain construction methods and properties of the universal integrals will be discussed in~our talk. Also, we will investigate convergence of universal integrals and provide new partial results.

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\noindent{\bf Keywords.} 
Universal integral, Sugeno integral, aggregation.

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\noindent{\bf Acknowledgement.}   
The present work was supported by the Internal grant VVGS-PF-2012-36.

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\bibitem{Boczek}
 Boczek M., Halčinová  L., Hutník O.,  Kaluszka M.: \emph{Novel survival functions based on conditional aggregation operators,} Inf. Sci. \textbf{580} (2021), 705--719.
\bibitem{Mesiar}
Klement E.P., Mesiar R., Pap E.: \emph{A universal integral as common frame for Choquet and Sugeno integral,} IEEE Trans. Fuzzy Systems \textbf{18}(1) (2010), 178--187.
\doi{10.1109/TFUZZ.2009.2039367}  
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