Why the Gauge (Henstock-Kurzweil) Integral Matters

The gauge integral (Henstock–Kurzweil)

The current (pure) mathematics curriculum at the university is well-established. Most of the choices made are sensible, but some important topics are still usually not taught. Some of these topics are obscure and not well known even to many professional mathematicians; others are known to specialists but for some reason are not touched upon in standard courses. The goal of this blog series is to visit some of these omissions, explain their basic theory, argue why they should be taught, and explore why they are omitted. Today’s topic is the gauge integral, also known as the Henstock–Kurzweil integral (also called the narrow Denjoy integral, the Luzin integral, or the Perron integral).

Riemann integral — two viewpoints

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 presented in essentially two equivalent ways.

First, the Darboux method bounds the “area” of $f$ from below and above. We take a partition $X = \{x_0,x_1,\dots,x_n\}$ with $x_0=a$, $x_n=b$ and $x_i\le x_{i+1}$. Then 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 is defined when $U_X(f) – L_X(f) \rightarrow 0$ as the partition $X$ gets finer. In that case the common limit of $U_X(f)$ and $L_X(f)$ is called the Darboux integral.

The second viewpoint uses Riemann sums. Again take partition points $X = \{x_0,\dots,x_n\}$ as above and choose sample points $\{a_0,\dots,a_{n-1}\}$ with $x_i\le a_i \le x_{i+1}$. The Riemann integral of $f$ is the limit of

$$\sum_{i=0}^{n-1} f(a_i)(x_{i+1} – x_i)$$

as the partition gets finer (mesh tends to zero). The two approaches are equivalent; see Bloch, Real Numbers and Real Analysis for a textbook treatment: https://www.amazon.com/Real-Numbers-Analysis/dp/0387721762

Extending the domain: improper integrals

One simple extension of the Riemann integral is to allow domains such as $[a,+\infty)$. For example, we define

$$\int_a^{+\infty} f(x)\,dx = \lim_{d\rightarrow +\infty} \int_a^d f(x)\,dx$$

when the limit exists. These improper Riemann integrals cover many important cases in applications.

Lebesgue integral and its advantages

Later in their courses, mathematics students learn the Lebesgue integral. The Lebesgue integral is built by partitioning the codomain of the function (values) rather than the domain, and this shift yields major advantages.

The Lebesgue integral is much more powerful in many respects: it integrates many more functions, it supports strong theorems that allow interchange of limits and integrals (dominated convergence, monotone convergence, etc.), and many proofs become cleaner. The setup of the Lebesgue theory is more technical than Riemann’s, but it pays off in generality and convenience.

Limitations of the Lebesgue integral

However, Lebesgue integration is not strictly stronger in every sense. While every Riemann-integrable function on a compact interval is Lebesgue integrable, some improper Riemann integrals are not Lebesgue integrable in the naive extended sense. For example,

$$\int_0^{+\infty}\frac{\sin(x)}{x}\,dx$$

is not Lebesgue integrable (over $[0,\infty)$) in the absolute sense, yet the improper Riemann integral exists and equals $\pi/2$. Thus the Lebesgue integral can be both more general and, in certain extended senses, weaker than improper Riemann integration.

Definition: the gauge integral

This tension is resolved by the gauge integral. The gauge integral (Henstock–Kurzweil) is a simple modification of the Riemann definition that produces a very powerful integral, one that encompasses the Riemann integral, many improper Riemann integrals, and the Lebesgue integral.

Recall a formal epsilon–delta style definition of the Riemann integral: we say the Riemann integral of $f:[a,b]\rightarrow\mathbb{R}$ is a number $L$ such that for every $\varepsilon>0$ there exists $\delta>0$ so that whenever a partition $x_0,\dots,x_n$ and sample points $a_1,\dots,a_n$ satisfy appropriate ordering 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 replaces the uniform $\delta$ by a positive function (a gauge). More precisely, the gauge integral of $f:[a,b]\rightarrow\mathbb{R}$ is a number $L$ such that for every $\varepsilon>0$ there exists a function $\delta:[a,b]\rightarrow(0,+\infty)$ with the property that whenever points $x_0,\dots,x_n$ and sample points $a_1,\dots,a_n$ satisfy the usual ordering and

$$x_i – x_{i-1}<\delta(a_i)\quad\text{for all }i,$$

then

$$\left|L – \sum_{i=1}^n f(a_i)(x_i – x_{i-1})\right|<\varepsilon.$$

Variants of this definition extend the domain $[a,b]$ in natural ways; those generalizations are not pursued here.

Key properties and examples

The gauge integral has several striking properties. First, a function $f$ is Lebesgue integrable if and only if it is gauge integrable and $\int |f|<+\infty$; thus Lebesgue integrable functions are precisely the absolutely gauge integrable functions. Intuitively, the Lebesgue integral corresponds to absolutely convergent series while the gauge integral corresponds to conditionally convergent series in the series analogy.

Second, many classical theorems for the Riemann and Lebesgue integrals hold in the gauge setting, often in cleaner and more general forms. For example, one classical statement of the fundamental theorem of calculus says: if $F$ is continuous on $[a,b]$ with derivative $F’=f$ and $f$ is Riemann integrable, then

$$\int_a^b f(x)\,dx = F(b)-F(a).$$

One drawback of that statement is the hypothesis “if $f$ is Riemann integrable.” The gauge integral removes that restriction: if $F’ = f$ everywhere on $[a,b]$ except possibly at countably many points, then $f$ is gauge integrable and

$$\int_a^b f(x)\,dx = F(b)-F(a).$$

Similarly, limit theorems, substitution rules, and integration-by-parts formulas admit strong, general versions in the gauge-integral framework.

Pedagogical advantages and limitations

Why should gauge integration receive more attention in undergraduate education? First, its definition is as easy to state as the Riemann integral and — with some practice — is intuitive. Second, it yields a single integral theory that unifies Riemann, many improper Riemann, and Lebesgue integrals on $\mathbb{R}$; from the gauge integral one can recover the Lebesgue integral as a special case. Third, many general theorems become cleaner and most general in the gauge setting, making it attractive from a conceptual standpoint.

Why isn’t it widely taught? The gauge integral’s main drawback is that it is naturally formulated for functions $\mathbb{R}\rightarrow\mathbb{R}$; although extensions exist for $\mathbb{R}^n$, the Lebesgue theory handles very general measure spaces and higher-dimensional domains more systematically. For applied scientists the Riemann integral is usually sufficient; for analysts the Lebesgue integral is indispensable. These pragmatic considerations help explain why gauge integration is often omitted in standard curricula.

Conclusion

Mathematics values the most beautiful and general statements for theorems. From that perspective the gauge integral is an important omission in undergraduate education: it is easy to state, unifies several common integrals, and yields very general, elegant theorems. What are your thoughts?

References

• Introduction to Real Analysis — Swartz & DePree. Uses the gauge integral; coverage is introductory. https://www.amazon.com/Introduction-Real-Analysis-John-DePree/dp/0471853917
• A Modern Theory of Integration — R. G. Bartle. Exhaustive treatment dedicated to the gauge integral. https://www.amazon.com/Modern-Integration-Graduate-Studies-Mathematics/dp/0821808451