The term “gauge function” has a few different definitions in math. Two you’re most likely to come across in calculus:
1. Gauge Function used in Gauge Integrals
In Gauge integration, a gauge function γ(τi) basically defines a locally finite partition which varies from point to point. Defined over a closed interval [a, b], it proves the existence of a γ-fine tagged partition of that interval.
Unlike Riemann integrals, which can be easily imagined as dividing any area into small chunks, gauge integrals are much more challenging to visualize.
“Since gauge functions are arbitrary, and the partitions must be made with the “correct” width, the existence of γ-fine partitions becomes abstract.” ~McInnis (2002)
Another way of defining a gauge function is:
γ(τi) = (τi – ∂(τi), (τi) + ∂(τi)),
Where:
∂ = a strictly positive function—one made up of strictly positive real numbers (i.e. real numbers greater than zero).
Defined in this way, the gauge function depends on the value of f at τ. Although this results in variable length intervals, this definition means that the Riemann integral (over the same interval) will have constant lengths (McInnis, 2002).
2. Minkowski Functional in Convex Sets
The gauge function was originally formulated for the study of self-similar sets (a class of fractals, where portions of the set resemble the bigger whole in some way). Gauge functions play an important role in the study of time change of Brownian motion (Kigami, 2019).
The gauge function, γC, also called the Minkowski functional of C, can be defined for a convex set (C) of E* as (Borwein & Vanderwerff, 2010)):
C ⊂ E γ(x) = inf{λ ≥ 0: x ∈ λ ∪}
*A convex set is basically where you can pass between any two points A and B in a set without every leaving the set. C of E means that the set is in Euclidean space, where the normal rules of Euclidean geometry apply.
References
Borwein, J. & Vanderwerff, J. (2010). Constructions, Characterizations and Counterexamples. E-book.
Eijnden, J. Fractal dimension of self-similar sets. Retrieved September 5, 2020 from: https://www.math.ru.nl/~mueger/THESES/Jesper_van_den_Eijnden_Bachelor_2018.pdf
Kigami, J. (2019). Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance. American Mathematical Society.
McInnis, E. (2002). Gauge Integration. Retrieved September 5, 2020 from:
https://apps.dtic.mil/dtic/tr/fulltext/u2/a407084.pdf