Who says you that a.e. continuous functions are Borel measurable? See here : math.stackexchange.com/a/3060677/778190 Completeness of the underlying measure space is needed to show that an a.e. continuous function is measurable. So, in particular every a.e. real valued continuous function on IR^n is Lebesgue measurable but not necessarily Borel measurable; Because Borel measure space is not complete. Sir, I earnestly request you that you please refrain from teaching something which is not known to you; In most of the cases students are misguided by those vague arguments. Pardon me please sir if I'm little rude.