Invariance of generalized quasiarithmetic means generated by different measures

In this paper, we investigate the invariance of the arithmetic mean with respect to generalized quasiarithmetic means, that is, solve the functional equation

$$\begin{aligned} & \left( \frac{f}{g}\right) ^{-1}\left( \frac{\int _0^1f(tx+(1-t)y)d\mu (t)}{\int _0^1g(tx+(1-t)y)d\mu (t)}\right) \\ & \quad + \left( \frac{h}{k}\right) ^{-1}\left( \frac{\int _0^1h(tx+(1-t)y)d\nu (t)}{\int _0^1k(tx+(1-t)y)d\nu (t)}\right) =x+y,\quad x,y \in I, \end{aligned}$$

where \(f,g,h,k:I\rightarrow {\mathbb {R}}\) are four continuous functions, g, k are positive, f/g, h/k are strictly monotone, and \(\mu , \nu \) are probability measures over the Borel sets of [0, 1].