On the commutation of generalized means on probability spaces

Let ff and gg be real-valued continuous injections defined on a non-empty real interval II, and let (X,ℒ,λ)(X,ℒ,λ) and (Y,ℳ,μ)(Y,ℳ,μ) be probability spaces in each of which there is at least one measurable set whose measure is strictly between 0 and 1.We say that (f,g)(f,g) is a (λ,μ)(λ,μ)-switch if, for every ℒ⊗ℳℒ⊗ℳ-measurable function h:X×Y→R for which h[X×Y]h[X×Y] is contained in a compact subset of II, it holds f−1(∫Xf(g−1(∫Yg∘hdμ))dλ)=g−1(∫Yg(f−1(∫Xf∘hdλ))dμ), where f−1f−1 is the inverse of the corestriction of ff to f[I]f[I], and similarly for g−1g−1.We prove that this notion is well-defined, by establishing that the above functional equation is well-posed (the equation can be interpreted as a permutation of generalized means and raised as a problem in the theory of decision making under uncertainty), and show that (f,g)(f,g) is a (λ,μ)(λ,μ)-switch if and only if f=ag+bf=ag+b for some a,b∈R, a≠0a≠0.