Iterated conditional expectations (Q968865)

The paper discusses two results of \textit{A. S. Cherny} and \textit{P. G. Grigoriev} [Finance Stoch. 11, No. 2, 291--298 (2007; Zbl 1144.91008)]. The first one is general and concerns the distance between arbitrary equidistributed bounded random variables on a nonatomic probability space. The second one is a consequence of the first one and establishes a law invariance theorem for \(L^{\infty}\) dilatation monotone maps (in the nonatomic setting). The authors give a straightforward analytic proof of the second result. This proof is motivated by a simple geometric idea. Then it is shown that the first result is implied by the second one. It is emphasized that both the theorems, when properly interpreted physically, have many counterintuitive consequences, not only in coherent risk theory, as in [Zbl 1144.91008].