Pettis integrability of Stone transforms

Letf be a bounded Pettis integrable function ranging in a Banach spaceX (the range of the indefinite Pettis integral is separable). We consider Pettis integrability conditions for the Stone transform off and relate this problem to the regular oscillation condition for the family of functions {x^*fx^*B(X^*)}, whereB(X^*) is the unit ball inX^*.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 238–253, August, 1996.