I am confused about lebesgue measure. I have heard that the lebesgue outer measure of the rational numbers is 0. So could someone please give an example of a set of open intervals such that: a. The union of these intervals contains the rational numbers on [0, 1] b. The sum of lengths of these intervals is less than 1. The definition of lebesgue outer measure implies that, if the lebesgue outer measure of the rationals is 0, then such a set of open intervals must exist. But I am flabberghasted as to how this could be so. Thanks very much in advance.