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Fubini type theorems for the BV integral. (English) Zbl 1151.28003

The R-integral (also known as the BV integral) is introduced by W. F. Pfeffer [Indiana Univ. Math. J. 40, No. 1, 259–270 (1991; Zbl 0747.26010)]. It is an extension of the Lebesgue integral, however, the Fubini theorem is false for the R-integral. So it is interesting to consider the following problems:
1. Assuming that the double integral and the iterated integrals exist, do they have the same value?
2. Assuming that the double integral exists and the iterated integrals are equal, is their value equal to that of the double integral?
In this paper, the authors show that if an R-integral defined over a compact interval \([a, b]\times [c,d]\subset\mathbb{R}^2\) satisfies certain conditions, then over any subinterval of \([a, b]\times[c, d]\), the iterated integral exists and is equal to the double integral. The last section is devoted to interesting counterexamples for the above problems 1 and 2.

MSC:

28A35 Measures and integrals in product spaces

Citations:

Zbl 0747.26010
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