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.
In , 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 , etc.
However, there are things in other than intervals that may be measured. For example, the rational numbers have measure 0 (or ). 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.)
In 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.
An Immeasurable Set
To obtain a set that is not measurable, consider the following equivalence relation: We say if . This partitions the interval [0,1) into uncountably many equivalence classes. Let denote a set obtained by selecting one element from each equivalence class (this requires the Axiom of Choice). Let be an enumeration of the rational numbers in in [0,1). Finally, let .
By construction of , the ‘s are disjoint and together form the whole interval . We know that each 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
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 is immeasurable.