Question: what is the difference between measure and outer measure?
Answer: This was a smart move, since it turns out that there is no nontrivial function satisfying 1-4 for every subset of R. Lebesgue outer measure (m*) is for all set E of real numbers where as Lebesgue measure (m) is only for the set the set of measurable set of real numbers even if both of them are set fuctions.