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In this chapter we introduce some integrals for real-valued maps with respect to Riesz space-valued set functions, which are not necessarily finitely additive, but in general can be simply only increasing. First we deal with the Sipos (symmetric) integral and prove the monotone and Lebesgue dominated convergence theorems, Fatou’s lemma and the submodular theorem.
Moreover we introduce the Choquet (asymmetric) integral, giving in particular some applications to the weak and strong laws of large numbers in the context of Riesz spaces.
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