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Summary:
For a new Perron-type integral a concept of convergence is introduced such that the limit $f$ of a sequence of integrable functions $f_k$, $ k \in \mathbb N$ is integrable and any integrable $f$ is the limit of a sequence of stepfunctions $g_k$, $ k \in \mathbb N$.
References:
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[2] J. Jarník and J. Kurzweil: Perron-type integration on $n$-dimensional intervals and its properties. Czechosl. Math. J. 45 (1995), 79–106. MR 1314532
[3] J. Kurzweil: Nichtabsolut konvergente Integrale. Teubner, Leipzig, 1980. MR 0597703 | Zbl 0441.28001
[4] E. J. McShane: A Riemann-type integral that includes Lebesgue-Stieltjes, Bochner and stochastic integrals. Mem. Amer. Math. Soc. 88 (1969), . MR 0265527 | Zbl 0188.35702
[5] J. Mawhin: Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields. Czechosl. Math. J. 31 (1981), 614–632. MR 0631606 | Zbl 0562.26004
[6] E. J. McShane: Unified Integration. Academic Press, 1983. MR 0740710 | Zbl 0551.28001
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