I just read proof from Royden of theorem: 'Every Monotonic functions are differentiable almost everywhere.' But proof use Vitali Covering Lemma. But Vitali Covering Lemma is based on fact if we assume Axiom of choice to be true. Is theorem 'Every monotonic function differentiable a.e.' is also dependent on AC or is there some other proof of this fact not depending on Axiom of choice.
Note: We are taking axiom of countable choice to be true.(We are not even allowing axiom of dependent choice)