Turing's unpublished algorithm for normal numbers

In an unpublished manuscript Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm for computing instances in this set. We complete his manuscript by giving full proofs and correcting minor errors. While doing this, we recreate Turing’s ideas as accurately as possible. One of his original lemmas remained unproved but we have replaced it with a weaker lemma that still allows us to maintain Turing’s proof idea and obtain his result.