A SCALAR VOLTERRA DERIVATIVE FOR THE PoU-INTEGRAL

In [8] and [9] J. Jarník and J.Kurzweil introduced an integration process (called PU-integral) for real valued functions on an interval of n with the use of suitably regularC-partitions of unity, instead of the usual partitions. The PU-integral is nonabsolutely convergent and in dimension one falls properly in between the Lebesgue and the Kurzweil-Henstock integrals. In [4], without assuming any regularity condition for the applied partition of unity, there is studied an integral (called the PoU-integral) for Banach valued functions defined on a σ-finite quasi-Radon measure space. In particular it is proved that it is equivalent to the generalized McShane integral as defined by Fremlin in [6]. Here we continue the investigation of the vector valued PoU-integrable functions started in [4]. In Section 3, using a form of the Henstock Lemma (see Proposition 1), we characterize the Pettis integrable functions which are also PoU-integrable by means of finite pseudopartitions (Theorem 1). In Section 4, using a suitable derivation base satisfying the strong Vitali covering condition, we prove the existence of a scalar form of the Volterra derivative of the