Acta Mathematica Hungarica

, Volume 153, Issue 2, pp 350–355

# Commutativity of integral quasi-arithmetic means on measure spaces

• D. Głazowska
• P. Leonetti
• J. Matkowski
• S. Tringali
Article

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