On the equivalence of the two existing extensions of the leximax criterion to the infinite case

Using a common framework, we consider the two existing extensions of the leximax criterion to infinite environments [Arlegi, R., Besada, M., Nieto, J., Vázquez, C., 2005. Freedom of choice: the leximax criterion in the infinite case. Mathematical Social Sciences 49, 1–15; Ballester, M., De Miguel, J.R., 2003. Extending an order to the power set: the leximax criterion. Social Choice and Welfare 21, 63–71], and show that, though the respective definitions of the rules and their axiomatic characterizations appear to differ considerably, they actually propose the same extension of the leximax criterion to the infinite case.