Real Analysis Exchange

A constructive definition of the n-dimensional ν(S)-integral in terms of Riemann sums

Dirk Jens F. Nonnenmacher

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Abstract

In a former paper ([Ju-No 1]) we introduced an axiomatic approach to the theory of non-absolutely convergent integrals in \(\mathbb{R}^n\). A specialization of this abstract concept leads to the well-behaved \(\nu (S)\)-integral over quite general sets \(A\) which yields the divergence theorem in its presently most general form. (See [Ju-No 3].) While the definition of the \(\nu (S)\)-integral is of descriptive type (i.e. in terms of an additive almost everywhere differentiable set function) we prove in this paper that it can equivalently be defined using Riemann sums. As an application we show that any function being variationally integrable over \(A\) in the sense of [Pf 3] is also \(\nu (S)\)-integrable over \(A\) and both integrals coincide.

Article information

Source
Real Anal. Exchange, Volume 21, Number 1 (1995), 216-235.

Dates
First available in Project Euclid: 3 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1341343237

Mathematical Reviews number (MathSciNet)
MR1377531

Zentralblatt MATH identifier
0870.26004

Subjects
Primary: 26A39: Denjoy and Perron integrals, other special integrals 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX] 26B20: Integral formulas (Stokes, Gauss, Green, etc.)

Keywords
non-absolutely convergent integral Riemann sums divergence theorem singularities

Citation

Nonnenmacher, Dirk Jens F. A constructive definition of the n -dimensional ν( S )-integral in terms of Riemann sums. Real Anal. Exchange 21 (1995), no. 1, 216--235. https://projecteuclid.org/euclid.rae/1341343237


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References

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