 # Commutativity of integral quasi-arithmetic means on measure spaces

Research paper by D. Głazowska, P. Leonetti; J. Matkowski; S. Tringali

Indexed on: 23 Dec '17Published on: 01 Dec '17Published in: Acta Mathematica Hungarica

#### 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 . 