Chalebgwa, Taboka PrinceMorris, Sidney A.2025-06-172025-06-172025-02Taboka Prince Chalebgwa, Sidney A. Morris. Number theoretic subsets of the real line of full or null measure. Electronic Research Archive, 2025, 33(2): 1037-1044. doi: 10.3934/era.2025046.2688-1594 (online)10.3934/era.2025046http://hdl.handle.net/2263/102853During a first or second course in number theory, students soon encounter several sets of "number theoretic interest". These include basic sets such as the rational numbers, algebraic numbers, transcendental numbers, and Liouville numbers, as well as more exotic sets such as the constructible numbers, normal numbers, computable numbers, badly approximable numbers, the Mahler sets S, T and U, and sets of irrationality exponent , among others. Those exposed to some measure theory soon make a curious observation regarding a common property seemingly shared by all these sets: each of the sets has Lebesgue measure equal to zero, or its complement has Lebesgue measure equal to zero. In this expository note, we explain this phenomenon.en© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0).Full measureNull measureLebesgue measureReal lineTranscendental numbersLiouville numbersArticle