Immeasurable Sets in R

The following example was brought up in two different classes that I am taking, within a couple days of each other. It is the classic example of an immeasurable set on the interval [0,1]. Now in math, words like measure and measurable have technical definitions, and lead to bizarre results like the Banach-Tarski paradox. I’m going to give a fairly informal explanation here.

Measure

In \mathbb{R}, the (Lebesgue) measure roughly indicates how long an interval is. For example, the measure of [0,1] is 1, and so is the measure of [2,3], or [40.5, 41.5], or [-6, -5]. The measure of [0,2] is 2, and so on. We would write it as m([3,8]) = 5, etc.

However, there are things in \mathbb{R} other than intervals that may be measured. For example, the rational numbers \mathbb{Q} have measure 0 (or m(\mathbb{Q}) = 0). In fact, the set of irrationals in [0,1] have measure 1 while the rationals in [0,1] have measure 0, so in this sense, almost all of the numbers in [0,1] are irrational. (Of course, this result is circular because the cardinality of the irrationals versus the rationals is what determined their measures in the first place. For a proof, see Cantor’s diagonal argument.)

Cantor setIn addition, the Cantor set (above) has measure 0, even though it has uncountably many points. On the other hand, the Smith-Volterra-Cantor set (below) has measure 1/2, even though it contains no intervals.

Smith-Volterra-Cantor_set

An Immeasurable Set

To obtain a set that is not measurable, consider the following equivalence relation: We say x \sim y if x - y \in \mathbb{Q}. This partitions the interval [0,1) into uncountably many equivalence classes. Let E denote a set obtained by selecting one element from each equivalence class (this requires the Axiom of Choice). Let \{r_n\} be an enumeration of the rational numbers in \mathbb{Q} in [0,1). Finally, let E_n = E + r_n \pmod 1.

By construction of E, the E_n‘s are disjoint and together form the whole interval [0,1). We know that each E_n has the same measure. Furthermore, we know the measure of [0,1) is 1, and the sum of the measures of disjoint sets is the measure of the union. Hence we have

\displaystyle \sum_{n=1}^{\infty} m(E_n) = \sum_{n=1}^{\infty} m(E_1) = 1,

which is impossible, as there is no such number which results in 1 after being added to itself an infinite number of times. Thus each of the sets E_n is immeasurable.

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