## 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

is satisfied by every \({\mathscr{L} \otimes \mathscr{M}}\)-measurable simple function \({h\colon X \times Y \to I}\) 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 [7].

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## References

G. Aumann: Vollkommene Funktionalmittel und gewisse Kegelschnitteigenschaften. J. Reine Angew. Math.,

**176**, 49–55 (1937)V. I. Bogachev,

*Measure Theory*, Vol. I, Springer-Verlag (2007).Chew S.H.: A generalization of the quasilinear mean with applications to the measurement of income inequality and decison theory resolving the Allais paradox. Econometrica,

**51**, 1065–1092 (1983)Daróczy Z., Maksa G., Páles Z.: Functional equations involving means and their Gauss composition. Proc. Amer. Math. Soc.,

**134**, 521–530 (2006)Kahlig P., Matkowski J.: On the composition of homogeneous quasi-arithmetic means. J. Math. Anal. Appl.,

**216**, 69–85 (1997)M. Kuczma,

*An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality*, Birkhäuser (Basel, 2009) (2nd edition).Leonetti P., Matkowski J., Tringali S.: On the commutation of generalized means on probability spaces. Indag. Math.,

**27**, 945–953 (2016)Maccheroni F., Marinacci M., Rustichini A.: Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica,

**74**, 1447–1498 (2006)J. Matkowski, Generalized weighted arithmetic means, in:

*Functional Equations in Mathematical Analysis*, T. M. Rassias and J. Brzdȩk (eds.), Springer Optim. Appl.**52**, Springer (New York, 2012), pp. 555–573.Matkowski J.: Lagrangian mean-type mappings for which the arithmetic mean is invariant. J. Math. Anal. Appl.,

**309**, 15–24 (2005)Ritt J.F.: Permutable rational functions. Trans. Amer. Math. Soc.,

**25**, 399–448 (1923)Strzalecki T.: Axiomatic foundations of multiplier preferences. Econometrica,

**79**, 47–73 (2011)

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P. L. was supported by a PhD scholarship from Università “Luigi Bocconi”.

S. T. was supported by the Austrian Science Fund (FWF), Project No. M 1900-N39.

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Głazowska, D., Leonetti, P., Matkowski, J. *et al.* Commutativity of integral quasi-arithmetic means on measure spaces.
*Acta Math. Hungar.* **153**, 350–355 (2017). https://doi.org/10.1007/s10474-017-0734-2

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DOI: https://doi.org/10.1007/s10474-017-0734-2