In this paper we observe that one of our main results in ‘Optimal transport and dynamics of circle expanding maps acting on measures’ [Ergod. Th. & Dynam. Sys.33(2) (2013), 529–548] has an interesting consequence: an infinitesimal version of the Furstenberg conjecture is false in a very strong way. More precisely, we find deformations of the Lebesgue measure on the circle which are first-order invariant simultaneously for all integer multiplications modulo 1. We also correct an error in a lemma of the mentioned article. Both the proof and the statement must be corrected, but the main results of the article are not affected.