## 0. Overview of Measure Theory

Table of Contents | 1. Measures ≫My very first exposure to the idea of the integral was through the Riemann integral. It was always explained as “the area under the curve”, and this motivated the intuitive construction of taking limits of Riemann sums.

In some ways, this construction functioned very well. For instance, it allowed us to establish the fundamental theorem of calculus:

**Theorem 1. The Fundamental Theorem of Calculus**

If \(f\) is continuous on \([a, b]\), then the function \(F(x)=\int_a^x f(t)\ dt\) is differentiable on \((a, b)\), and \[\frac{d}{dx}\int_a^xf(t)\ dt=f(x).\] If \(f\) is just Riemann integrable on \([a, b]\) and \(F\) is continuous on \([a, b]\) and differentiable on \((a, b)\) such that \(F'(x)=f(x)\), then we have that \[\int_a^bf(t)\ dt=F(b)-F(a).\]

In some other ways, however, this construction of the integral wasn’t particularly helpful, and it left some lingering questions about integration during the nineteenth century:

- Under what conditions can we interchange limits and integrals? \(\frac 1n\to 0\) pointwise (and even uniformly) on \(\R\), but \(\int\frac 1ndx\) does not always approach \(\int 0\ dx\). Either we need a new possibly stronger notion of convergence, or we need to impose additional constraints on the integrands.
- The Riemann sum does not generalise well beoynd \(\R\), so how can we generalise integrals to \(\R^n\)? What about integrating \(\R\)- or \(\mathbb C\)-valued functions on domains that aren’t Euclidean space? (Note that this last question is, in a way, addressed by differential geometry.)
- Let \(f(x)=0\) when \(x\) is irrational and \(f(x)=1\) when \(x\) is rational. On one hand, the Riemann integral \(\int_0^1 f(x)\ dx\) does not converge (good exercise!). On the other, the rationals are far smaller than the reals on \([0, 1]\): in a sense, one could say that \(f\) is “almost always” 0. Is there a way to assign meaningful values to integrals such as this one?

Henry Lebesgue developed measure theory and a new perspective on integration around the turn of the nineteenth century. This new perspective allowed mathematicians to extend the Riemann integral in a way that helped answer the above questions.

We’ll start with measure theory itself, which serves the same fundamental importance to Lebesgue integration that the theory of metric spaces does for Riemann integration. We’ll then move on to defining integration and apply this new theory to the convergence theorems. Finally, we’ll discuss how the Lebesgue integral remains consistent with the Riemann integral and how the Lebesgue integral can be swiftly extended to \(\R^n\) and other domains.

I am far from an expert on measure theory and real analysis. These notes will hence be primarily introductory, but I will still aim to give a thorough and rigorous discussion of the theory and mathematics. Hopefully you find these notes helpful on some level.