It’s Halloween! Which means that we here at Infinity Plus One get to do something spooky!
That’s right, we’re going to talk about sets of numbers so weird that even the very idea of length breaks when looking at them.
In the last few posts, we’ve been talking about how to measure the lengths of sets, even ones that are weird.
In How Long is Infinity?, we introduced how we measure things — by covering up a set with little intervals, and then calling the length of our set the smallest lengths of intervals that cover our set.
Using this length, it turns out that any countable set, like , or even the set of all rational numbers, has zero length.
In The Cantor Set, we showed that, though uncountable sets like [0,1] (all the numbers between 0 and 1) have positive length, there are uncountable sets with zero length. The Cantor set is the main example.
In this post, we want to look at non-measureable sets. Sets which are so weird that they break our “ruler” and make it impossible to make any sense at all of their length.
As a technical caveat, the “ruler” we’ve been discussing so far is technically the “outer Lebesgue measure,” which is not really the same as the “Lebesgue measure” that mathematicians use. However, the difference is buried in technical details that would distract from the story, so we’ll bury those important details for this post.
So what does an non-measureable set look like?
It’s gotta be weird. The definition of length, or measure, that we have is pretty robust. It can handle some pretty weird sets, like all the decimals with a 7 in them, and spit out a length.^{1}
So, in order to come up with a non-measureable set, we’re going to have to work hard.
What we’re going to do is come up with a weird set, and we’ll prove that if we add up infinitely many copies of it, somehow that total length will end up between 1 and 3. But that can’t be right, since adding up infinitely many of the same number always gives either 0 or infinity!^{2}
To start, we’re going to split all the real numbers into groups.
The first group is all the rational numbers, i.e., any number that can be written as a fraction of integers, like 2/3 or -712/2341. We’ll call this set , for “quotient.”
The other groups are all copies of the rational numbers, but shifted left or right by a different real number. For instance, we could have the group , which is the set of all numbers which are plus any rational number you want. Or you could have , which is the set of all numbers which are plus any rational number you want.
There are a whole lot of these groups, and every real number is in one of them. On the other hand, there is more than one way to name which set you’re talking about.
When we gave the two examples of these groups, we used and to define them. In other words, we picked a particular number, or , that happened to be in the group, and used it as a representative of that set.
But there’s more than one number in , and we could have used any of them to represent the set, not just . For instance, is the exact same set, with the exact same numbers in it. So is or …
…though would not be the same, since and do not differ by a rational number.
As long as our representatives differ from each other by a rational number, the sets are exactly the same.
Using these groups of numbers, we can now construct the unmeasureable set.
Let be the unmeasurable set. To construct it, look at each of the groups of numbers we came up with earlier. From each one, pick a single representative that happens to be between 0 and 1. For instance, from the set , we could pick the representative , which is between 0 and 1. From the set , we could pick 0 or 1 or 1/2 or any rational number between 0 and 1.
Now that we’ve chosen one representative from each group, it turns out that is unmeasurable!
Here’s how we’ll prove it.
Similar to how we took and moved all the numbers by to make , we can take our set and move all the numbers by a rational number , and make a new set .
To make things clearer, this means that if a number is in , then the number is in . And, in the opposite direction, if a number is in , that means that must have been in .
But there’s something funny about . No matter how small is, and never overlap!
Remember, each of the groups we came up with earlier had infinitely many different representatives we could have picked. But the representatives had to differ from each other by a rational number if they were supposed to represent the same group.
If and overlap, that means there would be a number in and . That means that would also have to be a number in . Thus there are two numbers ( and ) that are both in , but differ by a rational number.
But remember that the numbers in are representatives of our groups, and so if they differ by a rational number, they represented the same group.
But we only picked one representative from each group to put in .
And so and can’t overlap!
Next step: Put the rational numbers between -1 and 1 into some order. There are infinitely many of them, but they’re still a countable set, so we can do it. There are more details in the earlier post The size of infinity, but here’s the kind of ordering we could use to make sure we get them all.
Since we have an ordering for the rational numbers between -1 and 1, we’ll call the “1st” rational number, the “2nd,” etc., and the “k-th” rational number. Then, we can come up with a whole bunch of copies of moved around. We’ll call the the set , i.e., the set moved up or down by .
Just as before, none of the overlap. Also, since only had numbers between 0 and 1, and is between -1 and 1, then all the numbers in are between -1 and 2.
The more difficult part is to recognize that if you put all of the together (“take their union”), then together, they contain every number between 0 and 1.
To see this, pick any number between 0 and 1. No matter which number we pick, it’s in one of the groups we made earlier, perhaps . But, when we made , we picked one representative (between 0 and 1) for each of these groups. Since the representative and the number are in the same group, they have to differ by a rational number, and since they are both between 0 and 1, that rational number they differ by has to be between -1 and 1. That means that is in the set that happens to be moved by , which is a rational number!
Yeah, it’s kind of hard to keep all these sets straight, but we’re almost done.
To finally see that the set can’t be measured (i.e., is non-measurable), let’s pretend that we can measure it, and show that something impossible happens.
If we can measure , the sets have the same length, since they’re really the same set, just moved up or down on the number line.
Since, put together, the contain all the numbers between 0 and 1, their total length has to be at least 1. And, since the only have numbers between -1 and 2, clearly their total length has to be no bigger than 3. If we wrote to represent the length of , we could write that like this:
But, again, the set has the same length as each of the , and so, really, we’re saying:
But we’re adding up infinitely many of the same number! If the length were 0, adding up infinitely many zeros gives zero length. If the length were any number bigger than zero, adding up infinitely many of them would give infinite length!^{3}
And so, since 0 and are not between 1 and 3^{4}, we have shown something impossible. Thus cannot be measurable. We have broken our ruler.
So, yeah, non-measurable sets are weird. And we had to do a lot of work to come up with one.
But, in the end, it might seem like a waste of effort. I mean, it’s just a weird set that no one in their right mind would care about anyway.^{5}
But there are some weird things you can do with non-measurable sets.
The most famous is the Banach-Tarski paradox. There is a way you can take a sphere, cut it up into a few pieces, and rearrange them, and end up with two spheres, exactly identical to the original.
But that’s for next time!
Happy Halloween!
<– Previous Post: The Cantor Set
First post in this series: How Long is Infinity?
–> Next Post: Double for Nothing: the Banach-Tarski Paradox
- For all decimals between 0 and 1 that have a 7 in them, the length it gives is 1. In other words, virtually “every” number has a seven in it! ↩
- The name for the set we’ll be constructing is a “Vitali set.” You can read more on the Wikipedia page. ↩
- Importantly, there are only countably many sets and lengths that we’re considering, so there’s no funny business going on. Adding up countably many numbers or sets works the way you think it “should.” ↩
- Citation needed. ↩
- Of course, mathematicians are not really known for ever being in their right mind… ↩