# Commutativity of integral quasi-arithmetic means on measure spaces

Research paper by **Dorota Głazowska, Paolo Leonetti, Janusz Matkowski, Salvatore Tringali**

Indexed on: **11 Mar '17**Published on: **11 Mar '17**Published in: **arXiv - Mathematics - Classical Analysis and ODEs**

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#### Abstract

Let $(X, \mathscr{L}, \lambda)$ and $(Y, \mathscr{M}, \mu)$ be finite measure
spaces for which there exist $A \in \mathscr{L}$ and $B \in \mathscr{M}$ 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 $$
f^{-1}\!\left(\int_X f\!\left(g^{-1}\!\left(\int_Y g \circ
h\;d\mu\right)\right)d \lambda\right)\! = g^{-1}\!\left(\int_Y
g\!\left(f^{-1}\!\left(\int_X f \circ h\;d\lambda\right)\right)d \mu\right) $$
is satisfied by every $\mathscr{L} \otimes \mathscr{M}$-measurable simple
function $h: X \times Y \to I$ if and only if $f=c g$ 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].