1. Measures
≪ 0. Overview of Measure Theory | Table of Contents | 1.1. \(\sigma\)-Algebras ≫In order to develop the idea of a derivative and an integral, we needed to use the theory of metric spaces (as well as the completeness of \(\R\)).
In much the same way, in order to extend the integral, we need to first lay the groundwork with measures and measurable spaces.
From a physical or intuitive perspective, measures rigorously generalise our notions of length, area, weight, etc., just like how metrics rigorously generalise our notions of distance.
One particular property of metrics is the triangle inequality, and that comes from our intuitive knowledge of how distances work. If you want to travel from point A to point B, taking a detour won’t make that trip any shorter. In much the same way, we have some basic intuitive expectations of how measures should work. For now, I’ll use the “length” of an interval as our prototypical example: the length of \([a, b]\) is just \(b-a\) when \(b\geq a\).
- The length of the empty set is zero.
- If \(I_1\) and \(I_2\) are disjoint intervals, the length of \(I_1\cup I_2\) is the sum of the lengths of \(I_1\) and \(I_2\).
It helps to extend this second property (AKA disjoint additivity) to countable collections of disjoint sets, but this has the consequence of so-called unmeasurable sets. The Vitali set is an infamous example, and we’ll discuss this in more detail later in the chapter.
Measure theory ended up being highly applicable to probability theory, which lead to the development of things like stochastic calculus. This then blossomed into applied fields such as physics and economics, thanks to the concept of Brownian motion.
This chapter will focus specifically on measures and measurable spaces (sets with a measure defined on them). We’ll develop their basic properties and observe some helpful ways to classify measures.