Applications of anticompact sets to analogs of Denjoy–Young–Saks and Lebesgue theorems

We consider the problem of transfer of the Denjoy–Young–Saks theorem on derivates to infinite-dimensional Banach spaces and the problem of nondifferentiability of indefinite Pettis integral in infinite-dimensional Banach spaces. Our approach is based on the concept of an anticompact set proposed by us earlier. We prove an analog of the Denjoy–Young–Saks theorem on derivates in Banach spaces which have anticompact sets. Also in such spaces we obtain an analog of the Lebesgue theorem. This result states that each indefinite Pettis integral is differentiable almost everywhere in the topology of special Hilbert space generated by some anticompact set in the original space.