On the commutation of generalized means on probability spaces

Abstract

Let $f$ and $g$ be real-valued continuous injections defined on a non-empty real interval $I$, and let $(X,\mathcal{L},\lambda )$ and $(Y,\mathcal{M},\mu )$ 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 $(\lambda ,\mu )$-switch if, for every $\mathcal{L}\otimes \mathcal{M}$-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 $$f^{-1}\left({\int }_{X}f\left(g^{-1}\left({\int }_{Y}g\circ hd\mu \right)\right)d\lambda \right)=g^{-1}\left({\int }_{Y}g\left(f^{-1}\left({\int }_{X}f\circ hd\lambda \right)\right)d\mu \right),$$ 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 $(\lambda ,\mu )$-switch if and only if $f=ag+b$ for some $a,b\in \mathbf{R}$, $a\ne 0$.

9 pages, no figures. Fixed minor details. Final version to appear in Indagationes Mathematicae

References in corpus (1)

• Characterization of generalized quasi-arithmetic means

Cited by in corpus (2)

• On a functional equation related to two-variable weighted quasi-arithmetic means
• Commutativity of integral quasi-arithmetic means on measure spaces