We’re all vaguely familiar with the notion of length when it comes to intervals; the interval [1,2] (meaning the set of all numbers between 1 and 2 inclusive) has length 1, as does the interval (1,2) (the set of all numbers between 1 and 2, exclusive). Even though the latter excludes its endpoints, the points have length zero. So the question is, how do we measure more complicated sets? For example, what is the ‘length’ of the set of all rational numbers between 0 and 1? The intuitive way is this: suppose we have a set that we want to find the length of. Now, let T be a (countable) of intervals such that S is completely contained in the union of all the elements of T; mathematically speaking, we want

Since is contained in , we obviously should have that the length of is the length of , which is itself less than or equal to the sum of the lengths of its constituent intervals:

(where signifies the length of .
The fact that is countable is significant; if we allowed to be uncountable, then we could define as , for every and then every set would have length zero! Not exactly useful.
Obviously any will have many such s; so we can simply define to be the sum of the lengths of the smallest possible that covers ; or, if there is no smallest , the largest such that for all , there is a that covers such that . Again, written out, this becomes

This is what is known technically as the Lebesgue measure of a set, after Henri Lebesgue, the 19th century mathematician. This definition gives each set a Lebesgue measure and seems to behave normally under transformations such as translation; it’s not easy to construct a set that acts oddly under Lebesgue measure. But it is doable, and it was first found by a mathematician named Giuseppe Vitali. Let be the set of irrational numbers in , restricted so that if , then . Oh, and we also have to include 0 in . In other words, is a subset of such that no two members differ by a rational number. Then does not have a well-defined Lebesgue measure, and is called a Vitali set. To show this, first we order the rational numbers in the interval , . We can do this since the rational numbers are all countable. Then, we define

Each is disjoint from each other one, since if was in and , then we would have , for some or , an impossibility by the definition of . Let . Obviously, , but we also have To show this, pick some . If , then we’re done, else it was excluded since there is some such that . So we have
.
Now, consider what this implies for . We know that
. But since each is just moved around on the real line, we have , which will either be infinity or zero depending on whether V has zero measure or not. But by the above relation, we have

which directly contradicts what we derived about . So therefore, is not well-defined.

This naturally raises the question of how one can determine what sets can be assigned a measure. And here I have to turn to Wikipedia; according to them, what I’ve been calling the measure is actually the outer measure, and the actual measure, which I’ll call , is defined by
whenever for all sets
If this condition is not satisfied, then doesn’t have a measure. To see that fails this test, consider . Then obviously . Your homework for today is to prove that . It’s not enough to show that those two sets are uncountable; there exist uncountable sets of measure 0, such as the Cantor set.

One point that I glossed over in the construction of is that you have to somehow pick which irrational numbers you’re leaving out and which you’re leaving in. This requires what is known as the axiom of choice; the axiom of choice is independent of standard set theory, and mathematicians are divided on whether they work in set theory with or without the axiom of choice (though the majority do work with it). But that’s a topic for another time.