Analog of Denjoy-Young-Saks theorem about contingency for mappings into Fréchet spaces and one its application in theory of vector integration (Q2901781)

The present paper is devoted to the generalization of the Denjoy-Young-Saks theorem about derivative numbers to the case of mappings acting from a real interval \(I=[a,b]\) into Fréchet spaces. More precisely, it is well known for each real-valued function \(f\) acting from a real interval \(I=[a,b]\) that either \(f\) is differentiable almost everywhere or \(f\) has infinite derivative numbers almost everywhere on \(I\). For the case of mappings taking values in Fréchet spaces, an analogue of the notion of derivative number is introduced. It is proved that each separably-valued mapping \(f\) acting from \(I\) into Fréchet spaces has to satisfy one of the following conditions: either \(f\) is differentiable almost everywhere, or \(f\) has infinite derivative numbers almost everywhere on \(I\), or \(f\) does not have derivative numbers almost everywhere on \(I\). There is a mapping for which the last condition can be fulfilled everywhere on \(I\). Use of the above-mentioned result of the paper allows to prove a representation of each generalized (strongly) absolutely continuous and compact a.e. subdifferentiable mapping into an arbitrary LCS as a narrow Denjoy-Bochner integral. Also, a coincidence of the class of indefinite narrow Denjoy-Bochner integrals with generalized (strongly) absolutely continuous and a.e. differentiable mappings is obtained in the case of Fréchet spaces.