Henstock–Kurzweil Integral Overview

• The Henstock–Kurzweil integral is a generalized Riemann integral defined via pointwise gauges, allowing integration of functions that are nonabsolutely integrable.
• Its formulation using MC-antiderivatives unifies Riemann, Lebesgue, and Perron integrals, providing clear proofs for integration by parts and change-of-variables formulas.
• Extensions to multidimensional, vector, and fuzzy-valued functions, along with computational methods, broaden its impact in modern analysis and applied mathematics.

The Henstock–Kurzweil integral, also termed the gauge or generalized Riemann integral, is a nonabsolute integral that strictly extends both Riemann and Lebesgue integration by leveraging adaptive partitions controlled by pointwise gauges. It encompasses the integration of highly irregular and nonabsolutely integrable functions—particularly those arising as derivatives not captured by the Lebesgue theory—while maintaining a structure paralleling classical calculus. Formally introduced independently by Ralph Henstock and Jaroslav Kurzweil in the mid-20th century, the Henstock–Kurzweil (HK) integral has since been developed in various analytical, topological, and computational directions, with equivalence to established integrals such as Perron, Denjoy, and Stieltjes integrals, as well as encompassing multidimensional, vector- and set-valued, and time scale extensions.

1. Foundational Definition: Gauge Partitions and Integrability

The fundamental advancement of the Henstock–Kurzweil integral is the replacement of uniform or σ-additive mesh size control with a pointwise positive function called a gauge, denoted δ:[a,b] → (0,∞). A partition $\{([x_{i-1},x_i],\xi_i)\}_{i=1}^n$ of $[a,b]$ is δ-fine if each subinterval $[x_{i-1},x_i]$ contains its tag $\xi_i$ and satisfies $x_i - x_{i-1} < \delta(\xi_i)$. The HK integral of $f:[a,b] \to \mathbb{R}$ is defined as the number $A$ such that for every $\epsilon>0$, there is a gauge δ with the property that

$$\left|\sum_{i=1}^n f(\xi_i)(x_i - x_{i-1}) - A\right| < \epsilon$$

for every δ-fine partition. This definition, developed in (Henstock, 2017, Henstock et al., 2015), and (Bendová et al., 2010), admits local refinement near singularities of $f$ and supports the integration of all derivatives, even those that fail to be Lebesgue-integrable.

2. Structure Theorems, Equivalences, and MC-Antiderivatives

A refinement and simplification of the Perron integral is accomplished via the notion of MC-derivatives and MC-antiderivatives (Bendová et al., 2010). For $f$ to be the MC-derivative of $F$ on $I$, a strictly increasing control function $\varphi:I\rightarrow \mathbb{R}$ must satisfy for all $x\in I$: $$\lim_{y\to x}\frac{F(y)-F(x)-f(x)(y-x)}{\varphi(y)-\varphi(x)} = 0.$$ This shifts the complexity of multiple parameter controls (e.g., $\varepsilon$ and gauge in the classical HK theory) to a single monotonic $\varphi$. It is established that:

• The MC-integral is equivalent to the Perron and thus the Henstock–Kurzweil integral.
• Indefinite MC-integrals exist and are unique up to constants, generalizing Newton–Leibniz.
• The theory extends via “additive interval functions” to the multidimensional and Stieltjes settings; e.g., for additive set functions $F$ and $G$, and a superadditive $\Phi$, the condition

$$\lim_{k\to\infty}\frac{F(Q_k)-f(x)G(Q_k)}{\Phi(Q_k)} = 0$$

serves as a multidimensional analog (Bendová et al., 2010).

Thus, the MC-derivative formulation subsumes the classical, Riemann, Lebesgue, and gauge-based viewpoints and yields more transparent proofs for essential results (e.g., integration by parts, change-of-variables formulas), as in (Bendová et al., 2010).

3. Properties, Extensions, and Structural Results

Several properties of the Henstock–Kurzweil integral distinguish it within integration theory:

• Additivity and Linearity: As shown in (Henstock et al., 2015) and (Paxton, 2016), the HK integral is additive over intervals, linear, and monotonic.
• Fundamental Theorem of Calculus: The HK integral integrates every derivative, and conversely, its indefinite integral is differentiable almost everywhere recapturing the integrand (Henstock, 2017, Henstock et al., 2015).
• Compatibility with the Lebesgue Integral: If $f$ is Lebesgue integrable, then it is also Henstock–Kurzweil integrable with the same value (Henstock et al., 2015); reciprocally, a function is Lebesgue integrable iff both $f$ and $|f|$ are HK-integrable.
• Dominated/Monotonic Convergence: The HK integral admits monotone and dominated convergence theorems, formulated in terms of variation and control sets (Henstock et al., 2015).
• Sequential and Generalized Sequence Characterizations: Alternative definitions include convergence of Riemann sums under sequences of shrinking gauges, unifying ε–δ, Darboux, and topological forms (Paxton, 2016, Toneva, 2011).
• Extensions to Vector, Set, and Fuzzy-Valued Functions: The integral provides a natural theory in vector-valued (Kaliaj, 2018, Kaliaj, 2018, Kaliaj, 2019), Riesz space (You et al., 2017, Boccuto et al., 2011), and fuzzy (Zhao et al., 2017) contexts, supporting multidimensional, multifunction, and time-scale frameworks.

4. Descriptive and Completion Theories

The HK integral is not a Banach space with respect to natural norms (e.g., the Alexiewicz norm). Completions lead to the continuous primitive (distributional) integral (Talvila, 2019), in which $f$ is interpreted as a distributional derivative of a continuous “primitive” $F$, and integration over intervals or rectangles is computed via boundary values of $F$. Key properties include:

• The Banach space of primitives $B_c(\mathbb{R}^n)$ admits an isometric isomorphism with the space of integrable distributions $A_c(\mathbb{R}^n)$.
• The HK integrable functions are dense in $A_c$, and both $L^1$ and HK-integrable functions complete to this space.
• Rigorous results on duals, convolution, integration by parts, and Fubini-type theorems are naturally carried over (Talvila, 2019, Talvila, 28 Jan 2025).

5. Multidimensional, Abstract, and Computational Extensions

The multidimensional HK integral employs additive interval (set) functions, gauges on product spaces, and superadditive control functions. In practice, multidimensional integration is mediated via tagged partitions and interval functions, with Fubini-type and local-to-global patching theorems (Kaliaj, 2019, Kaliaj, 2018, Kaliaj, 2018). Time-scale calculus merges discrete and continuous theories under the HK integral for Riesz-space-valued functions (You et al., 2017).

Recent extensions (Edalat, 5 Mar 2025) generalize the HK construction to arbitrary compact metric spaces with respect to Borel measures, using net convergence of simple valuations over tagged partitions and gauges derived from measures way below the target Borel measure. This $D_\mu$-integral encompasses both bounded and unbounded functions and admits D$_\mu$-integrable functions that are not Lebesgue integrable, with explicit constructions in spaces such as the Cantor set.

Computationally, domain-theoretic methods using directed nets of simple valuations and valuations on partitions lead to frameworks for exact computation of integrals and solutions to differential equations without numerical error accumulation (Edalat, 5 Mar 2025). This approach supports “exact real arithmetic” and is crucial for applications in computable analysis.

6. Fourier and Fractional Calculus Integration

The HK integral enables representation of Fourier transforms for conditionally convergent integrals, extending classical inversion, convolution, and exchange results to broader function and distribution spaces (Talvila, 28 Jan 2025, Mahanta et al., 2022, Morales et al., 2020). In these extensions:

• The Fourier transform is rigorously defined as a second distributional derivative of an HK-based continuous primitive.
• The exchange theorem and inversion formulas hold in Alexiewicz norm and pointwise for $L^p$ classes.
• Sufficient regularity conditions yield bounded variation of Fourier transforms for $L^1$ functions in the HK context.
• Fractional calculus is extended to HK-integrable distributions, admitting fractional Riemann–Liouville operators as bounded operators, semigroup properties, and generalizations of Abel integral equations, with commensurate convolution and Fourier analysis (Morales et al., 2020).

7. Impact, Applications, and Generalizations

The HK integral’s adaptivity and flexibility underpin its utility in real analysis, measure theory, and applications:

• It serves as a canonical framework for integrating all derivatives while maintaining a transparent connection to the calculus fundamental theorem (Henstock et al., 2015, Henstock, 2017).
• Nonabsolutely convergent phenomena, such as the integration over currents, multifunctions, Riesz spaces, and fuzzy domains, are accommodated natively (Julia, 2019, Candeloro et al., 2016, Boccuto et al., 2011, Zhao et al., 2017).
• It is foundational for advanced integration in infinite-dimensional, dynamical, probabilistic, and computational models.
• Generalizations via domain theory, continuous primitive completions, and Laplace integral extensions facilitate further research in partial differential equations, harmonic analysis, and computable analysis (Edalat, 5 Mar 2025, Mahanta et al., 2022, Talvila, 2019).

The HK integral thus forms a unifying, robust, and extensible pillar in modern integration theory, linking classical and nonabsolute integration, and supporting ongoing development in abstract analysis, applied mathematics, and computation.