## 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 $$| F'|$$.

### MSC:

 26A39 Denjoy and Perron integrals, other special integrals 26B15 Integration of real functions of several variables: length, area, volume 28A12 Contents, measures, outer measures, capacities
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