McShane integral

A gauge integral using free tagged partitions whose tags may lie outside their associated subintervals, yielding for real-valued functions exactly the Lebesgue-integrable class.

Version

v1 · 2026-09-08 · History

Domain-specific #

5511

Origin domain

real analysis

Subdomain

gauge integration

Core Idea

A function is McShane integrable when for every epsilon one gauge makes all sufficiently fine free-tagged partition sums lie within epsilon of one value; unlike Henstock-Kurzweil partitions, tags need not belong to their subintervals. The variable gauge permits local control of interval width, while free tags enforce enough uniformity to recover absolute/Lebesgue integrability. In Banach-valued settings the construction relates to Bochner-type integration under suitable hypotheses. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application

McShane integral belongs to real analysis and is useful where the analyst can specify a function on a compact interval, a positive gauge, free tagged partitions, Riemann-type sums, and a candidate integral value, then evaluate every epsilon admits one positive gauge controlling all compatible free tagged partitions and their sums converge to the same value. The scope is broad within that domain but bounded by the need for every epsilon admits one positive gauge controlling all compatible free tagged partitions and their sums converge to the same value. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making every epsilon admits one positive gauge controlling all compatible free tagged partitions and their sums converge to the same value the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name McShane integral can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to McShane integral. McShane integral compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: a function on a compact interval, a positive gauge, free tagged partitions, Riemann-type sums, and a candidate integral value. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express every epsilon admits one positive gauge controlling all compatible free tagged partitions and their sums converge to the same value independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of real analysis because they reuse a function on a compact interval, a positive gauge, free tagged partitions, Riemann-type sums, and a candidate integral value, The variable gauge permits local control of interval width, while free tags enforce enough uniformity to recover absolute/Lebesgue integrability. In Banach-valued settings the construction relates to Bochner-type integration under suitable hypotheses., and type the carrier, state every parameter and convention in the definition, test that every epsilon admits one positive gauge controlling all compatible free tagged partitions and their sums converge to the same value, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction McShane integral Domain-specific

Parents (1) — more general patterns this builds on

• McShane integral is a kind of Aggregation Prime

  The proposed strict upward parent is prime:aggregation.

Hierarchy path (1) — routes to 1 parentless root

• McShane integral → Aggregation → Micro Macro Linkage

Neighborhood in Abstraction Space

McShane integral sits in a moderately populated region (53^rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Complex Analysis & Integral Transforms (29 abstractions)

Nearest neighbors

• Bochner integral — 0.89
• Locally integrable function — 0.89
• Banach–Mazur compactum — 0.88
• Cauchy's integral formula — 0.87
• Total variation — 0.87

Computed from structural-signature embeddings · 2026-09-08