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**A full descriptive definition of the Henstock-Kurzweil integral in the Euclidean space.**
*(English)*
Zbl 1047.26006

Lebesgue integrals are characterised by the fact that if \(\int_{{\mathbb R}^m}f\,d\lambda\) exists then we have absolute continuity of the indefinite integral, i.e., for every \(\varepsilon >0\) there is a \(\delta>0\) such that for every measurable set \(E\subset \mathbb R^m\) with \(\lambda (E)<\delta\) we have \(\int_E| f| \,d\lambda<\varepsilon\). On the real line the Henstock-Kurzweil integral is characterised by the fact that its indefinite integral, \(F(x):=\int_a^x f\), is an \(ACG_*\) function (generalised absolutely continuous in the restricted sense). In the paper under review, the author provides a characterisation of the Henstock-Kurzweil integral on a compact interval in \(\mathbb R^m\). This then solves a rather important problem in non-absolute integration. The paper includes a fairly complete review of the literature on this problem.

The main result is the following. Theorem 4.3. { Let \(F\) be an additive interval function defined on all subintervals of a given compact interval \(E\subset\mathbb R^m\). The following conditions are equivalent:

i) \(F\) is the indefinite Henstock-Kurzweil integral of some function on \(E\);

ii) the variational measure \(V_{{\mathcal HK}}F\) is absolutely continuous.}

The variational measure of \(X\subset \mathbb R^m\) is defined by \[ V_{{\mathcal HK}}F(X)=\inf_\delta \sup_{\mathcal P} \sum_{i=1}^p| F(I_i)| . \] The infimum is over all gauge functions \(\delta \!:\!X\to(0,\infty)\). The supremum is over all \(\delta\)-fine partitions \({\mathcal P}=\{(I_i, \xi_i)\}_{i=1}^p\), i.e., for all \(1\leq i\leq p\), \(\xi\in I_i\) and closed interval \(I_i\), is a subset of the open ball with centre \(\xi_i\) and radius \(\delta(\xi_i)\). Here the partition is anchored on \(X\), i.e., \(\xi_i\subset X\) for all \(1\leq i\leq p\). And, \(V_{{\mathcal HK}}F\) is absolutely continuous if \(V_{{\mathcal HK}}F(X)=0\) whenever the Lebesgue measure of \(X\) is zero.

It is shown that condition ii) can be replaced with the strong Lusin condition: For each set \(Z\subset E\) of Lebesgue measure zero and all \(\varepsilon>0\) there is a gauge \(\delta\) on \(Z\) such that \(\sum_{i=1}^p| F(I_i)| <\varepsilon\) for each \(\delta\)-fine partition \({\mathcal P}=\{(I_i, \xi_i)\}_{i=1}^p\) anchored on \(Z\). It is also shown that \(V_{{\mathcal HK}}F(X)<\infty\) if and only if \(F\) is the indefinite integral of an \(L^1\) function. As well as being absolutely continuous, the variational measure \(V_{{\mathcal HK}}F\) is \(\sigma\)-finite and its Radon-Nikodým derivative with respect to Lebesgue measure is equal to \(| F'| \).

The main result is the following. Theorem 4.3. { Let \(F\) be an additive interval function defined on all subintervals of a given compact interval \(E\subset\mathbb R^m\). The following conditions are equivalent:

i) \(F\) is the indefinite Henstock-Kurzweil integral of some function on \(E\);

ii) the variational measure \(V_{{\mathcal HK}}F\) is absolutely continuous.}

The variational measure of \(X\subset \mathbb R^m\) is defined by \[ V_{{\mathcal HK}}F(X)=\inf_\delta \sup_{\mathcal P} \sum_{i=1}^p| F(I_i)| . \] The infimum is over all gauge functions \(\delta \!:\!X\to(0,\infty)\). The supremum is over all \(\delta\)-fine partitions \({\mathcal P}=\{(I_i, \xi_i)\}_{i=1}^p\), i.e., for all \(1\leq i\leq p\), \(\xi\in I_i\) and closed interval \(I_i\), is a subset of the open ball with centre \(\xi_i\) and radius \(\delta(\xi_i)\). Here the partition is anchored on \(X\), i.e., \(\xi_i\subset X\) for all \(1\leq i\leq p\). And, \(V_{{\mathcal HK}}F\) is absolutely continuous if \(V_{{\mathcal HK}}F(X)=0\) whenever the Lebesgue measure of \(X\) is zero.

It is shown that condition ii) can be replaced with the strong Lusin condition: For each set \(Z\subset E\) of Lebesgue measure zero and all \(\varepsilon>0\) there is a gauge \(\delta\) on \(Z\) such that \(\sum_{i=1}^p| F(I_i)| <\varepsilon\) for each \(\delta\)-fine partition \({\mathcal P}=\{(I_i, \xi_i)\}_{i=1}^p\) anchored on \(Z\). It is also shown that \(V_{{\mathcal HK}}F(X)<\infty\) if and only if \(F\) is the indefinite integral of an \(L^1\) function. As well as being absolutely continuous, the variational measure \(V_{{\mathcal HK}}F\) is \(\sigma\)-finite and its Radon-Nikodým derivative with respect to Lebesgue measure is equal to \(| F'| \).

Reviewer: Erik O. Talvila (Chilliwack)