McShane integral

Given a closed interval [a, b] of the real line, a free tagged partition ${\displaystyle P}$ of ${\displaystyle [a,b]}$ is a set

${\displaystyle \{(t_{i},[a_{i-1},a_{i}]):1\leq i\leq n\}}$

where

${\displaystyle a=a_{0}<a_{1}<\dots <a_{n}=b}$

and each tag ${\displaystyle t_{i}\in [a,b]}$.

The fact that the tags are allowed to be outside the subintervals is why the partition is called free. It's also the only difference between the definitions of the Henstock-Kurzweil integral and the McShane integral.

For a function ${\displaystyle f:[a,b]\to \mathbb {R} }$ and a free tagged partition ${\displaystyle P}$, define ${\displaystyle S(f,P)=\sum _{i=1}^{n}f(t_{i})(a_{i}-a_{i-1}).}$

A positive function ${\displaystyle \delta$ :[a,b]\to (0,+\infty )} is called a gauge in this context.

We say that a free tagged partition ${\displaystyle P}$ is ${\displaystyle \delta }$-fine if for all ${\displaystyle i=1,2,\dots ,n,}$

${\displaystyle [a_{i-1},a_{i}]\subseteq [t_{i}-\delta (t_{i}),t_{i}+\delta (t_{i})].}$

Intuitively, the gauge controls the widths of the subintervals. Like with the Henstock-Kurzweil integral, this provides flexibility (especially near problematic points) not given by the Riemann integral.

The value ${\displaystyle \int _{a}^{b}f}$ is the McShane integral of ${\displaystyle f:[a,b]\to \mathbb {R} }$ if for every ${\displaystyle \varepsilon >0}$ we can find a gauge ${\displaystyle \delta }$ such that for all ${\displaystyle \delta }$-fine free tagged partitions ${\displaystyle P}$ of ${\displaystyle [a,b]}$,

${\displaystyle \left|\int _{a}^{b}f-S(f,P)\right|<\varepsilon .}$