Decomposition approaches to integration without a measure

TL;DR

This paper introduces a generalized integration framework based on decomposition systems with weighting functions, encompassing economic optimization, linear programming, and combinatorial problems, and proposes new integrals related to optimization.

Contribution

It extends the approach of Even and Lehrer by generalizing decomposition-based integrals and introduces new types of integrals tailored for optimization tasks.

Findings

Generalizes existing decomposition-based integrals

Connects integrals with economic and optimization problems

Proposes new integrals for optimization applications

Abstract

Extending the idea of Even and Lehrer [3], we discuss a general approach to integration based on a given decomposition system equipped with a weighting function, and a decomposition of the integrated function. We distinguish two type of decompositions: sub-decomposition based integrals (in economics linked with optimization problems to maximize the possible profit) and super-decomposition based integrals (linked with costs minimization). We provide several examples (both theoretical and realistic) to stress that our approach generalizes that of Even and Lehrer [3] and also covers problems of linear programming and combinatorial optimization. Finally, we introduce some new types of integrals related to optimization tasks.