The concept of `negligible variation' (Q1978827)

The article is devoted to the generalized Riemann (= Kurzweil-Henstock) integral. Let \(E\) be a subset of \(I:=[ a,b ]\). A subpartition \(\{[ u_i,v_i ]\}_{i=1}^s\) of \(I\) tagged by \(t_i\in [ u_i,v_i ]\) is \((\delta , E)\)-fine for a gauge function \(\delta : E\rightarrow (0,\infty)\) if \([ u_i,v_i]\subset [ t_i-\delta (t_i),t_i+\delta (t_i) ]\), \(i=1,\ldots ,s\). A function \(F:[ a,b ]\rightarrow \mathbb R\) is said to have negligible variation (NV) on a set \(E\subset [ a,b ]\) if for every \(\varepsilon >0\) there exists a gauge function \(\delta _\varepsilon \) on \(E\) such that for every \((\delta _\varepsilon , E)\)-fine subpartition \(\{([ u_i,v_i ],t_i)\}\) we have \[ \sum _{i=1}^s |F(v_i)-F(u_i) |<\varepsilon \;. \] A typical result: A function \(G:I\rightarrow \mathbb R\) is an indefinite integral of some generalized Riemann integrable function \(f:I\rightarrow \mathbb R\) if and only if there exists a null set \(Z\subset I\) such that \(G'(x)=f(x)\) for all \(x\in I\setminus Z\), and \(G\) is of NV on \(Z\). Then we have \(G(x)-G(a)=\int _a^x f\) for all \(x\in I\). The article contains a sketch of the proof and also a limit theorem based on NV.