A discriminant of the primitive of a K-H integrable function in Euclidean space (Q1430487)

For a finitely additive interval function \(F\) defined on an \(m\)-dimensional interval \(E\) the quantity \(D_cF(x)\) is defined as \(\lim_{| I_c| \to 0}\frac{F(I_c)}{| I_c| }\) if the limit exists and as \(0\) otherwise. (\(I_c\) is a cubic interval for which \(x\in E\) is one of its vertices.) Using \(D_cF\) a characterization of the property that \(F\) is the primitive of a Henstock-Kurzweil integrable (real) function \(f\) is presented. For a given finitely additive \(F\) and any point-interval pair \((x,I)\), \(x\in E\), \(I\subset E\) define \[ H(x,I)=(| F(I)| +| I| )\cdot\frac{| \frac{F(I)}{| I| }-D_cF(x)| }{| \frac{F(I)}{| I| }-D_cF(x)| +1} \] and set \( \mu(F) = \inf_{\delta}\sup_{\delta \text{ fine }D} \sum_{D} H(x,I)\), the discriminant (\(\delta\) is a gauge on \(E\), \(D\) a partial division \(\{(x,I)\}\), \(x\) one of the vertices of \(I\)). It is proved that a given finitely additive \(F\) is a primitive of a Henstock-Kurzweil integrable (real) function \(f\) if and only if \(\mu(F)=0\).