Acta Mathematica Hungarica

, Volume 153, Issue 2, pp 350–355 | Cite as

Commutativity of integral quasi-arithmetic means on measure spaces

  • D. Głazowska
  • P. Leonetti
  • J. Matkowski
  • S. TringaliEmail author


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
$$f^{-1}\Big(\int_X f\Big(g^{-1}\Big(\int_Y g \circ h \,d\mu\Big)\Big)d \lambda\Big) = g^{-1}\Big(\int_Y g\Big(f^{-1}\Big(\int_X f \circ h \,d\lambda\Big)\Big)d \mu\Big)$$
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].

Key words and phrases

functional equation commuting mapping generalized (quasi-arithmetic) mean 

Mathematics Subject Classification

primary 26E60 39B22 39B52 secondary 28E99 60B99 91B99 


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  1. 1.
    G. Aumann: Vollkommene Funktionalmittel und gewisse Kegelschnitteigenschaften. J. Reine Angew. Math., 176, 49–55 (1937)MathSciNetzbMATHGoogle Scholar
  2. 2.
    V. I. Bogachev, Measure Theory, Vol. I, Springer-Verlag (2007).Google Scholar
  3. 3.
    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)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Daróczy Z., Maksa G., Páles Z.: Functional equations involving means and their Gauss composition. Proc. Amer. Math. Soc., 134, 521–530 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Kahlig P., Matkowski J.: On the composition of homogeneous quasi-arithmetic means. J. Math. Anal. Appl., 216, 69–85 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality, Birkhäuser (Basel, 2009) (2nd edition).Google Scholar
  7. 7.
    Leonetti P., Matkowski J., Tringali S.: On the commutation of generalized means on probability spaces. Indag. Math., 27, 945–953 (2016)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Maccheroni F., Marinacci M., Rustichini A.: Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica, 74, 1447–1498 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    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.Google Scholar
  10. 10.
    Matkowski J.: Lagrangian mean-type mappings for which the arithmetic mean is invariant. J. Math. Anal. Appl., 309, 15–24 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Ritt J.F.: Permutable rational functions. Trans. Amer. Math. Soc., 25, 399–448 (1923)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Strzalecki T.: Axiomatic foundations of multiplier preferences. Econometrica, 79, 47–73 (2011)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  • D. Głazowska
    • 1
  • P. Leonetti
    • 2
  • J. Matkowski
    • 1
  • S. Tringali
    • 3
    Email author
  1. 1.Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona GóraZielona GóraPoland
  2. 2.Department of StatisticsUniversità “Luigi Bocconi”MilanoItaly
  3. 3.Institute for Mathematics and Scientific ComputingUniversity of Graz, NAWI GrazGrazAustria

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