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.
- \(\{1, 2, 3\}\). 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.
- \(\mathbb{R}\). This denotes the set of all ‘real’ numbers (hence the R): in other words, the number line.
- \(\{x \in \mathbb{R} : 0 \leq x\leq 1\}\). 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, \(1/\pi \),…).
- \(\{\text{red, orange, yellow, green, blue, indigo, violet}\}\). 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.
- 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.
