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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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

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

First available in Project Euclid: 3 July 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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.)

non-absolutely convergent integral Riemann sums divergence theorem singularities


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.

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