The proof given follows some ideas by John C. Oxtoby. For positive integers "n" > 2 and "q" ≥ 2 set:  we have  Observe that for each positive integer "n" ≥ 2 and "m" ≥ 1, we also have  Since  and "n" > 2 we have  Now  and it follows that for each positive integer "m", "L" ∩ (−"m", "m") has Lebesgue measure zero. Consequently, so has "L". In contrast, the Lebesgue measure of the set of "all" real transcendental numbers is infinite (since the set of algebraic numbers is a null set). For each positive integer "n", set  The set of all Liouville numbers can thus be written as  Each "Un" is an open set; as its closure contains all rationals (the formula_37 from each punctured interval), it is also a dense subset of real line. Since it is the intersection of countably many such open dense sets, "L" is comeagre, that is to say, it is a "dense" Gδ set. The Liouville-Roth irrationality measure (irrationality exponent, approximation exponent, or Liouville–Roth constant) of a real number "x" is a measure of how "closely" it can be approximated by rationals. Generalizing the definition of Liouville numbers, instead of allowing any "n" in the power of "q", we find the largest possible value for "μ" such that formula_38 is satisfied by an infinite number of integer pairs ("p", "q") with "q" > 0. This maximum value of "μ" is defined to be the irrationality measure of "x". For any value "μ" less than this upper bound, the infinite set of all rationals "p"/"q" satisfying the above inequality yield an approximation of "x". Conversely, if "μ" is greater than the upper bound, then there are at most finitely many ("p", "q") with "q" > 0 that satisfy the inequality; thus, the opposite inequality holds for all larger values of "q". In other words, given the irrationality measure "μ" of a real number "x", whenever a rational approximation "x" ≅ "p"/"q", "p","q" ∈ N yields "n" + 1 exact decimal digits, we have  for any ε>0, except for at most a finite number of "lucky" pairs ("p", "q").