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Minimization interchange theorem on posets 1 1 Introduction 1 2 Minimization interchange theorem on posets 2 2.1 Main result 2 2.2 A sufficient condition for Φ-directed sets 6 3 Applications to minimization on functional spaces 7 3.1 Interchange between minimization and integration 7 3.1.1 Main result with integrals 7 3.1.2 Comparison with Giner [7] 9 3.1.3 Comparison with Rockafellar and Wets [13] 9 3.2 Comparison with Shapiro [14] 11 3.3 Interchange between minimization and Choquet’s integral 13 4 Conclusion 14 Appendix A Extended Lebesgue and outer integrals 14 A.1 Functional space L0(Ω,F,μ;R) and the Lebesgue integral 15 A.2 Functional spaces L1⊕(Ω,F,μ;R), L1⊖(Ω,F,μ;R) and the extended Lebesgue integral 16 A.3 Outer integral on L0(Ω,F,μ;R) 19 References 21
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