Subdiagonal and Almost Uniform Distributions

Abstract

A distribution (function) $F$ on $[0,1]$ with $F(t)$ less or equal $t$ for all $t$ is called subdiagonal. The extreme subdiagonal distributions are identified as those whose distribution functions are almost surely the identity, or equivalently for which $F \circ F = F$. There exists a close connection to exchangeable random orders on $\{1,2,3,...\}$.