A gould type integral with respect to a multisubmeasure

In [GOULD, G. G.: Integration over vector-valued measures, Proc. London Math. Soc. (3) 15, (1965), 193–205], G. G. Gould introduced a type of integral of a bounded, real valued function with respect to a finite additive set function taking values in a Banach space, integral which is more general than the Lebesgue one. Recently, A. Precupanu and A. Croitoru gave the generalization, defining a Gould type integral for multimeasures with values in _kc(X), X being a Banach space ([PRECUPANU, A.—CROITORU, A.: A Gould type integral with respect to a multimeasure, I, An. Ştiinţ. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. 48 (2002), 165–200]). Taking as starting point this work and [PRECUPANU, A.—CROITORU, A.: A Gould type integral with respect to a multimeasure, II, An. Ştiinţ. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. 49 (2003), 183–207], we define here the notion of a Gould type integral with respect to a _bf(X)-valued multisubmeasure, pointing out important properties of it. We also establish that, even if we deal with multisubmeasures, the integral is still a multimeasure.