# Commutativity of integral quasi-arithmetic means on measure spaces

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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

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

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].

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## Author information

Authors

### Corresponding author

Correspondence to S. Tringali.

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