## Real Analysis Exchange

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

Dirk Jens F. Nonnenmacher

#### 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

https://projecteuclid.org/euclid.rae/1341343237

Mathematical Reviews number (MathSciNet)
MR1377531

Zentralblatt MATH identifier
0870.26004

#### 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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