Commutativity of integral quasi-arithmetic means on measure spaces

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].