Comparison between different types of abstract integrals in Riesz spaces. (English) Zbl 0881.28008

Boccuto, Antonio; Sambucini, Anna Rita

Rend. Circ. Mat. Palermo, II. Ser. 46, No. 2, 255-278 (1997).

Let \(E\) be a Dedekind complete Riesz space and \([a,b]\) a real interval. In the first part of the paper, the authors give a characterization of “Riemann-integrable” functions \(f:[a,b]\to E\) well known in the real-valued case. In the second part, they introduce an integral for real-valued functions with respect to an \(E\)-valued measure: A function \(f\) is integrable if there is an \(L^1\)-Cauchy sequence of simple functions \((o)\)-converging in measure to \(f\). This concept of integrability is compared with several other concepts of integrability.

Reviewer: H.Weber (Udine)

MSC:

28B15

Set functions, measures and integrals with values in ordered spaces

Keywords:

Dedekind complete Riesz space; concept of integrability

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