On the Vietoris semicontinuity property of the $L_p$ balls at $p=1$ and an application

Abstract

In this paper, the Vietoris right lower semicontinuity at $p=1$ of the set valued map $p\rightarrow B_{\Omega ,\mathcal {X},p}(r)$, $p\in [1,\infty ]$, is discussed where $B_{\Omega ,\mathcal {X},p}(r)$ is the closed ball of the space $L_{p}(\Omega ,\Sigma ,\mu ; \mathcal {X})$ centered at the origin with radius r, $(\Omega ,\Sigma ,\mu )$ is a finite and positive measure space, $\mathcal {X}$ is a separable Banach space. It is proved that the considered set valued map is Vietoris right lower semicontinuous at $p=1$. Introducing additional geometric constraints on the functions from the ball $B_{\Omega ,\mathcal {X},1}(r)$, a property, which is close to the Hausdorff right lower semicontinuity, is derived. An application of the obtained result to the set of integrable outputs of the input–output system described by a Urysohn type integral operator is studied.