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arXiv · 1703.03938

Commutativity of integral quasi-arithmetic means on measure spaces

Dorota Głazowska·Paolo Leonetti·Janusz Matkowski·Salvatore Tringali
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Abstract

Let $(X, \mathscr{L}, λ)$ and $(Y, \mathscr{M}, μ)$ be finite measure spaces for which there exist $A \in \mathscr{L}$ and $B \in \mathscr{M}$ with $0 < λ(A) < λ(X)$ and $0 < μ(B) < μ(Y)$, 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 $$ f^{-1}\!\left(\int_X f\!\left(g^{-1}\!\left(\int_Y g \circ h\;dμ\right)\right)d λ\right)\! = g^{-1}\!\left(\int_Y g\!\left(f^{-1}\!\left(\int_X f \circ h\;dλ\right)\right)d μ\right)$$ 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 $λ(X)=μ(Y)=1$, was recently obtained in [Indag. Math. 27 (2016), 945-953].

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Dorota Głazowska, Paolo Leonetti, Janusz Matkowski, Salvatore Tringali. 2017-04-12. Commutativity of integral quasi-arithmetic means on measure spaces. https://doi.org/10.1007/s10474-017-0734-2

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