Commutativity of integral quasi-arithmetic means on measure spaces

Abstract

Let $(X,\mathcal{L},\lambda )$ and $(Y,\mathcal{M},\mu )$ be finite measure spaces for which there exist $A\in \mathcal{L}$ and $B\in \mathcal{M}$ with $0<\lambda (A)<\lambda (X)$ and $0<\mu (B)<\mu (Y)$, and let $I\subseteq \mathbf{R}$ be a non-empty interval. We prove that, if $f$ and $g$ are continuous bijections $I\to {\mathbf{R}}^{+}$, then the equation $$f^{-1}\left({\int }_{X}f\left(g^{-1}\left({\int }_{Y}g\circ hd\mu \right)\right)d\lambda \right)=g^{-1}\left({\int }_{Y}g\left(f^{-1}\left({\int }_{X}f\circ hd\lambda \right)\right)d\mu \right)$$ is satisfied by every $\mathcal{L}\otimes \mathcal{M}$-measurable simple function $h:X\times Y\to I$ if and only if $f=cg$ for some $c\in {\mathbf{R}}^{+}$ (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 $I\to \mathbf{R}$ and $\lambda (X)=\mu (Y)=1$, was recently obtained in [Indag. Math. 27 (2016), 945-953].

5 pages, no figures. To appear in Acta Mathematica Hungarica. The paper is a sequel of arXiv:1503.01139

References in corpus (1)

• On the commutation of generalized means on probability spaces