Superpositional measurability of a multivalued function under generalized Сaratheodory conditions

I. D. Serova

Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 135, pp. 305-314

Résumé

For a multivalued mapping $F:[a,b]\times \mathbb{R}^{m}\to \mathrm{comp}(\mathbb{R}^{n})$, the problem of superpositional measurability and superpositional selectivity is considered. As it is known, for superpositional measurability it is sufficient that the mapping $ F $ satisfies the Caratheodory conditions, for superpositional selectivity it is sufficient that $ F (\cdot, x) $ has a measurable section and $F(t, \cdot)$ is upper semicontinuous. In this paper, we propose generalizations of these conditions based on the replacement, in the definitions of continuity and semicontinuity, of the limit of the sequence of coordinates of points in the images of multivalued mappings to a one-sided limit. It is shown that under such weakened conditions the multivalued mapping $ F $ possesses the required properties of superpositional measurability / superpositional selectivity. Illustrative examples are given, as well as examples of the significance of the proposed conditions. For single-valued mappings, the proposed conditions coincide with the generalized Caratheodory conditions proposed by I.V. Shragin (see [Bulletin of the Tambov University. Series: natural and technical sciences, 2014, 19:2, 476–478]).

Détail

Keywords: the Caratheodory condition, the Nemytsky multivalued operator, superpositional measurability, superpositional selectivity.