The Riemann integral is the first, naive integral one builds when learning to do calculus. Philosophically, it reflects the key notion in calculus: that of linearising continuous objects. (There is something to be said here about how continuity is a geometric, or topological notion, while linearity is an algebraic notion, but this is one of those things that are exceptionally illegal for someone at my level of experience to blather on about).

The technical procedure of the Riemann integal is straightforward: given a bounded function \(f: [a,b] \rightarrow \mathbb{R}\), one partitions the domain \([a,b]\) into finitely many pieces, and considers a piecewise constant approximation of \(f\), with the pieces determined by the partition. As the pieces get smaller, we recover \(f\) more precisely, and thus, with faith in the formalism of limiting procedures, we happily claim to have the (Riemann) integral of \(f\).

The more sceptical of us actually check when this formalism works, and we discover the conditions for Riemann integrability, which are somewhat tedious to write down. The amount of intuition they provide is rather low—there is no simple concept we can infer from them. This is already strike one against the Riemann integral.

Perhaps counterexamples will help. For instance, we learn early on that the Dirichlet function

\[f(x) = \begin{cases} 1 & x \in \mathbb{Q} \\ 0 & x \in \mathbb{Q}^c \end{cases},\]

is not Riemann integrable, because the choice of approximating constant on each piece inevitably leads to multiple possible values for \(\int f\). In fact, lots of functions are not Riemann integrable. Possibly too many. That’s strike two.

We haven’t gotten to the worst offense yet. One extremely desirable property in an integral is the existence of nice theorems regarding the following question. Given a sequence of functions \(f_n\), that converge at each point of the domain to a function \(f\), can we say that \(\int f_n\) converges to \(\int f\)? This is desirable in particular because it tells us that the integral is doing its job. An integral should collate a large quantity of local data into a single global output. It is a tool to talk about the big picture. But even more importantly, an integral should be capable of doing this in a “continuous” manner as often as possible. We would like to see, without necessarily resorting to very strict conditions, the classical notion of continuous behaviour: taking the output of a limit to the limit of the outputs.

The Riemann integral does a bad job of this. This sequence of functions, called “moving boxes”,

converges pointwise to \(0\) (“almost everywhere”): at each point on the \(x\)-axis (except \(x=0\)), once the box passes the point, it never returns. But notice that the integral of each box is always \(1\): because the height is \(n\) and the width is \(\frac{1}{n}\). So the functions converge to \(0\) but the integrals converge to \(1\). Clearly, pointwise convergence of functions is not enough for the Riemann integral.

What is enough then? The answer is uniform convergence. This is a significantly stronger notion, and thus applies in far fewer situations. This is the really big problem with the Riemann integral—strike three, out.

Who’s in next?

Lebesgue.

In his 1902 PhD thesis, Henri Lebesgue developed the Lebesgue integral to remedy all these problems. It provides a simple formulation for Riemann integrability, it applies to a larger class of functions, and it has some very nice convergence theorems that apply under suitably weak conditions. Talk about a dissertation.

In the Lebesgue philosophy, we partition the range of the function, rather than the domain. Thus, one needs a way to measure the chunks of the domain that map into each piece of the partitioned range. This led to the development of measure theory. In the next post, I’ll look at the “simple formulation for Riemann integrability” alluded to, and try to probe its limits.