There is a well-known structure theorem for locally compact non discrete topological division algebras, see here

https://math.stackexchange.com/q/1160086/187521

(I repost it here because I think it is more suitable given the nature of the question) and the proof of this theorem generally always uses the existence of Haar measures on locally compact topological groups. Intuitively I don't see how we could go without this existence, but I would like to be sure. *Could we prove the structure theorem without Haar measure or not ?*