The inversion of approximate and dyadic derivatives using an extension of the Henstock integral (Q757626)

A distribution S on [a,b] is a collection of measurable sets \(\{S_ x:\;x\in [a,b]\}\) such that \(x\in S_ x\) and x is a point of density of \(S_ x\). A collection \(\{(x_ i,[c_ i,d_ i]):\;i=1,...,N\}\) of tagged intervals is called a partition, which is S-subordinate to a positive function \(\delta\) if and only if \(\{[c_ i,d_ i]:\;i=1,...,N\}\) is a partition of [a,b], \(d_ i-c_ i<\delta (x_ i)\) and \(x_ i\in [c_ i,d_ i]\) and \(c_ i,d_ i\in S_{x_ i}\) for each i. Using this notion the author has introduced the Henstock-type integral and has proved basic properties of this integral, including S- differentiability almost everywhere. The second part of the paper is devoted to the study of so called dyadic Henstock integral which is defined with the use of families of intervals with dyadic end-points.