Omissions in Mathematics Education: Gauge Integration

The current (pure) mathematics curriculum at the university is well-established. Most of the choices made are sensible. But still, some important topics are usually not taught. Some of these topics are very obscure and not even very well known to many professional mathematicians, others are known to experts but for some reason are not touched upon in education. The goal of this blog series is to visit some of these omissions, explaining their basic theory, explaining why they should be taught, and finding out why they aren’t. Today’s topic is the gauge integral, also known as the Henstock-Kurzweil integral, the narrow Denjoy integral, the Luzin integral, or the Perron integral.

Every mathematics student is of course acquainted with the Riemann integral. The Riemann integral is defined for functions $f:[a,b]\rightarrow \mathbb{R}$. The Riemann-integral of $f$ can be defined in essentially two ways. First, we have the Darboux method. The Darboux method involves bounding the “area” of $f$ by below and by above. Indeed, we exhibit some numbers $X = \{x_0,x_1,…,x_n\}$ with $x_0= a$, $x_n = b$ and $x_i\leq x_{i+1}$. Then we define
$$U_X(f) = \sum_{i=0}^{n-1} (x_{i+1} – x_i)\sup_{x\in [x_i,x_{i+1}]}f(x)$$
and
$$L_X(f) = \sum_{i=0}^{n-1} (x_{i+1} – x_i)\inf_{x\in [x_i,x_{i+1}]}f(x)$$

The Darboux integral can then be defined if $U_X(f) – L_X(f) \rightarrow 0$ if $X$ gets finer. The limit of $U_X(f)$ (and $L_X(f)$) is then called the Darboux integral.

The second way of defining the Riemann integral involves again some numbers $X = \{x_0,x_1,…,x_n\}$ like above and also a set of points $\{a_0,…,a_{n-1}$ with $x_i\leq a_i\leq x_{i+1}$. The Riemann integral of $f$ can then be defined as the limit of
$$\sum_{i=0}^{n-1} f(a_i)(x_{i+1} – x_i)$$
as $X$ gets finer.

Both approaches are equivalent, and the resulting integral is denoted as $\int_a^b f(x)dx$. For the equivalence of both approaches see Bloch https://www.amazon.com/Real-Numbers-Analysis/dp/0387721762

An extension of the Riemann integral can be achieved by extending the domain to intervals like $[a,+\infty)$. Indeed, we can define
$$\int_a^{+\infty} f(x)dx = \lim_{d\rightarrow +\infty} \int_a^d f(x)dx$$

Later in their courses, mathematics students replace the Riemann integral with the Lebesgue integral. The Lebesgue integral is defined not by splitting the domain into a partition, but rather by splitting the codomain into a partition.

This results in a much more powerful integral. It can find the integral to many more functions, it satisfies many good properties that allow interchange of integral and limits, and the proofs involving Lebesgue integration are much less cumbersome. Only setting up the Lebesgue integral is very tedious.

But not all is perfect. While any Riemann integrable function $f:[a,b]\rightarrow \mathbb{R}$ is Lebesgue integrable, the same is not true for the extended Lebesgue integral. Indeed, the integral
$$\int_0^{+\infty}\frac{\sin(x)}{x}dx$$
is not Lebesgue integrable, but the Riemann integral is $\pi/2$. So the Lebesgue integral is at the same time more powerful, but also weaker than the Riemann integral.

This is solved by the gauge integral. The gauge integral is a very easy modification of the Riemann integral but yields a very powerful integral that encompasses the Riemann integral, the extended Riemann integral, and the Lebesgue integral. Let us first revisit the formal definition of the Riemann integral. We say that the Riemann integral of $f:[a,b]\rightarrow \mathbb{R}$ is a number $L$ such that for all $\varepsilon >0$, there exists a $\delta>0$ such that whenever $x_0,…,x_n$ and $a_1,…,a_n$ are numbers such that
$$a=x_0\leq a_1\leq x_1\leq a_2\leq …\leq x_{n-1}\leq a_n\leq x_n = x$$
and $x_i – x_{i-1}<\delta$ for all $i$, then
$$\left|L – \sum_{i=1}^n f(a_i)(x_i – x_{i-1})\right|<\varepsilon$$
The gauge integral is the exact same definition except that we replace the number $\delta$ by a function $\delta$. We get that the gauge integral of $f:[a,b]\rightarrow \mathbb{R}$ is a number $L$ such that for all $\varepsilon >0$, there exists a function $\delta:[a,b]\rightarrow (0,+\infty)$ such that whenever $x_0,…,x_n$ and $a_1,…,a_n$ are numbers such that
$$a=x_0\leq a_1\leq x_1\leq a_2\leq …\leq x_{n-1}\leq a_n\leq x_n = x$$
and $x_i – x_{i-1}<\delta(a_i)$ for all $i$, then
$$\left|L – \sum_{i=1}^n f(a_i)(x_i – x_{i-1})\right|<\varepsilon$$

Remark that similar definitions can be made that extend the domain $[a,b]$ to more general domains, but this is not pursued here.

The two definitions look similar, but there is a world of difference. The gauge integrals yield the most perfect possible integral for functions $\mathbb{R}\rightarrow \mathbb{R}$. Let us find out why by exhibiting some selected results.

First, a function $f$ is Lebesgue integrable if and only if it is gauge integrable and $\int |f|<+\infty$. So we retain the Lebesgue integral as a special case. Indeed, the Lebesgue integral is analogous to an absolutely convergent series, while the gauge integral is analogous to the convergent series in general. Furthermore, integrals like $\int_0^{+\infty}\frac{\sin(x)}{x}dx$ that can not be found by Lebesgue, can be found by the gauge integral and yield sensible answers.

Second, many theorems in Lebesgue integration and Riemann integral are special cases of gauge integral theorems. Furthermore, the theorems often take on much more beautiful forms in the gauge integral setting. One example is the fundamental theorem of calculus. Indeed, the second part of the fundamental theorem of calculus is as follows: Let $f$ and $F$ be functions $[a,b]\rightarrow \mathbb{R}$ such that $F$ is continuous and $F’ = f$. If $f$ is Riemann integrable, then
$$\int_a^b f(x)dx = F(b) – F(a)$$

One of the main drawbacks of this theorem is the part “If $f$ is Riemann integrable”. Can’t we deduce Riemann’s integrability from the fact that it is a derivative? It’s not possible. The gauge integral gives us a much much more general statement however (note that there are even more powerful forms): If $F$ and $f$ are functions $[a,b]\rightarrow \mathbb{R}$ such that $F’ = f$ for all points on $[a,b]$ except possibly countably many, then $f$ is gauge integrable and $\int_a^b f(x)dx = F(b) – F(a)$.

Other classical theorems like limit theorems, substitution theorems, and integration by parts theorems can all be generalized to hold in great generality.

So why should gauge integration receive more attention in undergrad education? First, the gauge integral is as easy to define as the Riemann integral and – with some experience – is very intuitive. Second, we get a more general integral encompassing all kinds of integrals mathematicians will see eventually. Using the gauge integral, it is possible to construct the Lebesgue integral as a special case. Third, we get a lot of general theorems which encompass both the Lebesgue as the Riemann cases. These theorems are the most general possible and are not as easily proved (if it is even possible) in the context of Lebesgue or Riemann integration.

So why isn’t the gauge integral taught? The gauge integral has one major drawback: it can only be defined for functions $\mathbb{R}\rightarrow \mathbb{R}$, although modifications are found to extend it to multivariable domains $\mathbb{R}^n$. The Lebesgue integral however accepts very very general domains. So the Lebesgue integral is still a fundamental part of knowledge even if the gauge integral is available on $\mathbb{R}$. Furthermore, the greater generality of the gauge integral is rarely needed for practicing scientists (who have enough with the slightly more intuitive Riemann integral), but also not for practicing mathematicians who would exclusively use the Lebesgue integral.

Still, I would say that mathematics is a lot about finding the most beautiful and general statements for theorems. And for that reason alone, the gauge integral is an important omission in undergraduate education. What are your thoughts?

For people interested in learning the gauge integral, I can provide the following two references:
– Introduction to Real Analysis by Swartz and Depree https://www.amazon.com/Introduction-Real-Analysis-John-DePree/dp/0471853917 This is a basic course on real analysis that uses the gauge integral. The covering of the gauge integral is rather superficial, however, but the most important results are there.
– A modern theory of integration by Bartle https://www.amazon.com/Modern-Integration-Graduate-Studies-Mathematics/dp/0821808451 This gives a very exhaustive treatment of the gauge integral. The entire book is dedicated to it, and the theory is explained in full detail. It contains everything you’ll ever want to know about this topic (and more!).