Interpolation of level sets for equimeasurable functions (Q1269582)

Let \((T,{\mathcal F},\mu)\) be a measure space with a \(\sigma\)-algebra \({\mathcal F}\) and a positive measure \(\mu\). Two \({\mathcal F}\)-measurable functions \(f\) and \(g\) are called rearrangements of each other if, for every real number \(\alpha\), one has \[ \mu(f^{-1} ([\alpha,+\infty)))= \mu(g^{-1}([\alpha, +\infty))). \] If \(R(f)\) denotes the family of rearrangements of a given measurable function \(f\), the author proves that, for each finite family \(f_1,\dots, f_n\) of equimeasurable functions in \(R(f)\), it is possible to construct a family of equimeasurable functions \((\eta_\lambda)_{\lambda\in[0,1]}\) in \(R(f)\) which interpolates the functions \(f_1,\dots, f_n\) in a convexlike way. As an application, this interpolation result yields a compact fixed point property.