Interchange of vector valued integrals when the measures are Bochner or Pettis indefinite integrals

Abstract

Necessary and sufficient conditions for the interchange of two Bochner integrals and for the interchange of two Pettis integrals are obtained. These conditions are different from those generally required in classical Fubini theorems since they do not require the construction of the cross product measure. The proof makes use of the Vitali-Hahn-Saks Theorem. It should be noted that while Fubini theorems use the cross product measure, one of the difficulties encountered is that the product measure fails to be countable additive – this is pointed out in M. Bhaskara Rao (Indiana Univ. Math. J. 21 (1972), 847–848) and Charles Swartz (Bull. Austral. Math. Soc. 8 (1973), 359–366). Most applications require the interchange of the two integrals rather than integration with respect to the product measure.