Uniform convergence theorem for the \(H_1\)-integral revisited (Q1417175)

A function \(f: [a,b]\to\mathbb{R}\) is \(H_1\)-integrable on \([a,b]\) to a number \(A\) if there exists a positive function \(\delta\) on \([a,b]\) such that for every \(\varepsilon> 0\) there exists a division \(D_0\) of \([a,b]\) such that \(|(D)\sum f(\xi)(v-u)- A|\leq\varepsilon\) for every \(\delta\)-fine division \(D\supseteq D_0\). Here \(D\supseteq D_0\) if for every \(([s,t],\eta)\in D\) there is \(([u,v],\xi)\in D_0\) such that \([s,t]\subset [u,v]\). It is proved in this note that the uniform convergence theorem does not hold for the \(H_1\)-integral.