Commutativity of Integral Quasi-Arithmetic Means on Measure Spaces

Let (X, L, λ) and (Y, M, μ) be finite measure spaces for which there exist A∈ L and B∈ M with 0 < λ(A) < λ(X) and 0 < μ(B) < μ(Y) , and let I⊆ R be a non-empty interval. We prove that, if f and g are continuous bijections I→ R+, then the equation (Formula Presented.) is satisfied by every L⊗ M-measurable simple function h: X× Y→ I if and only if f = cg for some c∈ R+ (it is easy to see that the equation is well posed). An analogous, but essentially different result, with f and g replaced by continuous injections I→ R and λ(X) = μ(Y) = 1 , was recently obtained in [7].