On the commutation of generalized means on probability spaces

Let $f$ and $g$ be real-valued continuous injections defined on a non-empty real interval $I$, and let $(X, \mathscr{L}, λ)$ and $(Y, \mathscr{M}, μ)$ 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 $(f,g)$ is a $(λ, μ)$-switch if, for every $\mathscr{L} \otimes \mathscr{M}$-measurable function $h: X \times Y \to \mathbf{R}$ for which $h[X\times Y]$ is contained in a compact subset of $I$, it holds $f^{-1}\!\left(\int_X f\!\left(g^{-1}\!\left(\int_Y g \circ h\;dμ\right)\right)d λ\right)\! = g^{-1}\!\left(\int_Y g\!\left(f^{-1}\!\left(\int_X f \circ h\;dλ\right)\right)d μ\right)\!,$ where $f^{-1}$ is the inverse of the corestriction of $f$ to $f[I]$, and similarly for $g^{-1}$. 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 $(f,g)$ is a $(λ, μ)$-switch if and only if $f = ag + b$ for some $a,b \in \mathbf R$, $a \ne 0$.

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