A Nonmeasurable Set from Coin Flips

To motivate the elaborate machinery of measure theory, it is desirable to have an example of a set which is not measurable in some natural space. The usual example is the Vitali set, obtained by picking one representative from each equivalence class of R induced by the relation x ∼ y iff x − y ∈ Q. The translation-invariance of Lebesgue measure implies that the resulting set is not Lebesgue-measurable [4]. By the Solovay Theorem [3], one cannot construct such a set in ZermeloFrankel set theory without appealing to the axiom of choice. In this note we give a variant construction in the language of probability theory, using the axiom of choice in the guise of the well-ordering principle [5]. For other constructions see [2, Ch. 5]. Consider the measure space (Ω,F , P), where Ω = {0, 1}, and F is the product σ-algebra, and P is the product measure ( 1 2 δ0 + 1 2 δ1) . This is the probability space for a sequence of independent fair coin flips indexed by Z. It is well-known that (Ω,F , P) is isomorphic up to null sets to Lebesgue measure on [0, 1), via binary expansion [1, Theorem 3.19].