Surer than almost sure: Measure theory

Measure theory

In this post and the following one, I will explain a bugbear of mine: the fact that something is described as ‘almost surely’ true, when it should really just be described as ‘surely’ true. The area of maths this falls in is ‘axiomatic’ probability theory, which is probability theory viewed as a sub-branch of something called measure theory, and so the thrust of this first post will be to briefly introduce measure theory.

Sets

Measure theory is, you guessed it, all about measuring things. The things you can measure are ‘sets’: a set is just the fancy way of saying any collection of things. Here are some sets.

1. \(\{1, 2, 3\}\). The curly brackets or braces are the way mathematicians denote sets, so this set has the numbers 1, 2 and 3 in it. The fancy name for the things in a set are the ‘elements’ of the set, so the elements of \(\{1, 2, 3\}\) are 1, 2 and 3.
2. \(\mathbb{R}\). This denotes the set of all ‘real’ numbers (hence the R): in other words, the number line.
3. \(\{x \in \mathbb{R} : 0 \leq x\leq  1\}\). I’ve introduced another way mathematicians write down sets: this means the collection of all real numbers (hence the \(x \in \mathbb{R} \) bit) which satisfy the condition \(0 \leq x \leq 1 \). So this is just the portion of the number line between zero and one (including for example 1/2, 0.23, \(1/\pi \),…).
4. \(\{\text{red, orange, yellow, green, blue, indigo, violet}\}\). Sets don’t have to consist of obviously mathematical things like numbers, but can include whatever you like, like here the colours of the rainbow. You could even mix and match, so \(\{1,  \text{red}\}\) is a set.
5. The set of all points on the ground which lie within Paris. I choose Paris partly because it’s well defined as a city: inside the ‘périphérique’ you’re in Paris, and outside you’re not.

 

Measures

 

The main object of interest in measure theory are measures which record how big things are. There are lots of different measures, because there are lots of different ways to decide how big something is. Here are some different ways you could measure the size of Paris (and other cities).

• The geographical area
• The volume, drawing straight lines up from the ground to the edge of the earth’s atmosphere. This will give a measure which is roughly just a constant times the geographical area, but not quite: some parts of the world are much closer to space than others 
• The number of people who live there
• The number of people who were there at 2am on Thursday 6 February 2020 (this is the type of measure that censuses like to use, because it is easier to define than ‘who lives there’ and ensures no-one gets counted twice)
• The number of people on the electoral roll for the city mayoral election
• The total amount of wealth owned by the city. Like with censuses, to turn this into a proper mathematical measure would require pinning down the definition, but you could in principle do that.

I hope that’s shown that measures really do do just what they sound like they should: measure things. Here’s some properties all measures have in common (if something doesn’t have one of these properties, mathematicians won’t call it a measure):

1. No set has negative measure (the ‘wealth’ example could fall foul of this if you tried to include debt)
2. To measure some things combined you can measure separately and add (the number of people living in the UK is the number living in England plus the number in Scotland plus the number in Wales plus the number in Northern Ireland; this wouldn’t be true if you counted students in their home and student areas, again highlighting the need for care in something like the census).

And that’s it! If you have an example satisfying both conditions, it’s a measure. (We have to be a little careful about how many pieces you can combine in the second condition. I’ll discuss this more in my next post.) Here’s one more example (closely related to the ‘number of people’ measure).

Counting measure: This just counts how many things are in the set. So the size of \(\{1, 2, 3\}\) would be 3, the size of \(\{5, 10\}\) would be 2, and the size of \(\{x\in \mathbb{R} : 0 \leq x \leq 1\}\) would be infinite: there are infinitely many numbers between zero and one (for example \(1/2, 1/3, 1/4, 1/5…\) are all in this set).

Like in the above example, the measure of a set is even allowed to be infinite. Another example of a set having infinite measure would be if you tried to define the area of a 3d shape (rather than its volume). To allow things to have infinite measure and the definition to still work well, mathematicians add a third condition to the two I gave: if you measure nothing you get 0 out. This is automatically true from the second condition unless every set you’re measuring is infinitely big.

 

Slicing sets up

 

What’s the volume of a circle? Not the area (which is given by the formula \(A=\pi r^2\)) but the volume? A proper mathematical circle (unlike ones you actually see in real life) has no thickness. The formula for the volume of a cylinder of thickness \(h\) and radius \(r\) is \[V= \pi r^2 h.\] If we plug in a thickness of zero we get the volume of a circle: zero. This means we can’t find the volume of a cylinder by chopping it into infinitely many circles and adding up their volumes. Remember I said we needed to be careful about how many pieces you can combine by adding up their sizes? This is why!
In the careful mathematical definition of a measure you are allowed to chop into infinitely many pieces sometimes. For example, given a cylinder of thickness 1, we could chop into a piece of thickness \(1/2\), another of thickness \(1/4\), another of thickness \(1/8\) and so on. We would be allowed to add their volumes together – and we’d get the volume of the original cylinder – even though there are infinitely many. But to make up our cylinder of thickness 1, it takes too many cylinders of thickness 0 (i.e. circles) for this to be allowed, even though it’s also infinitely many. Yes maths is weird!

 

Almost everywhere

 

You might think a problem with the above is that we shouldn’t chop into pieces with zero volume. But we can do that without problems if there aren’t too many. For example, we could slice into two pieces: one circle with no volume, and the other having all of the volume of the cylinder. (I know which piece of cake I’d choose!)
We would then have one piece which makes up essentially none of the cylinder, and another which makes up essentially all of it. The zero volume piece we call a ‘null set’. We say that ‘almost everywhere’ in the cylinder is in the other piece. I’ll use ‘almost every’ and similar terms to mean what they sound like in relation to almost everywhere (so ‘almost every’ element of the cylinder is in the other piece).
Here are a couple more examples which can hopefully help your intuition.

• The whole numbers \(\{1, 2, 3, …\}\) are a null set of real numbers (with respect to lengths), so almost every real number is not an integer. (Imagine throwing a dart at a number line. There’s no chance it would land exactly on an integer: even if it looks like it at first, as you zoom in you’ll see it’s at 1.000000001 or 3.9999997 or something).
• When you measure the size of an area by the number of people on the electoral roll, places that comprise only non-voters (under 18s, migrants who don’t have voting rights, or people who just haven’t bothered/been able to register) are null sets, and a set would have ‘almost everyone’ in it if it included every voter. Politically speaking, the number of voters who care about an issue is the right measure when you’re looking to win power. The fact that some groups of people end up being null sets then helps fuel problems like the fact that issues affecting young people are a smaller focus than those affecting older people. Who thought measure theory had no impact on the real world?!