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


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].