Cartan's 1940 paper, Sur la mesure de Haar, claims to provide a proof of the existence and uniqueness of Haar measure for locally compact abelian groups without the use of axiom of choice. However, Cartan claimed the existence of a partition of unity, citing Dieudonné's paper which ultimately uses Urysohn's lemma.
Similarly, in Alfsen's exposition of a simplified version of Cartan's argument, Alfsen also claimed the existence of a partition of unity without elaboration.
Given that Urysohn's lemma is not provable without choice, my questions are 1) can we construct partitions of unity on locally compact spaces without invoking Urysohn's lemma or any form of choice? 2) Do we truly have a choice-free proof of the existence and uniqueness of Haar measure for locally compact abelian groups?
