Comments, Follow-up work, Updates and Corrections

As much as software gets patched and updated from time to time, papers and theories develop and undergo an evolution, extensions and occasionally also corrections. As a big fan of scientific ethics and the Honkong principles on research integrity, I list hereafter changes, updates and corrections that I either find myself or get told by others (which I also appreciate and acknowledge).

Errare humanum est, and may anyone cast the first stone, who is without sin. This is what escaped my eyes, as well as those of my collaborators and also the hard-working people doing scientific peer reviews:

• Lemma 2 and Theorem 2 both need the additional hypothesis of \(f_1(a)\neq f_2(a)\) at \(a=\max(\Omega)\) and with \(\Omega\) being the union of both supports. (Thanks to Vincent Bürgin for pointing this out by providing a respective example). Since this condition is satisfiable by truncating the distributions at any smaller value \(a-\varepsilon\) for some \(\varepsilon>0\), Theorem 1 only asserts the invariance w.r.t. the ultrafilter on a dense subset of the whole space of probability distributions. See also the respectively updated preprint on arxiv for more details.