Completeness of the space of absolutely and upper integrable functions with values in a semi-normed space

Abstract

This paper studies absolute integrability for functions with values in semi- normed spaces and in locally convex topological vector spaces (LCTVS). We introduce an \emph{upper-integral} approach (based on a $\rho$-variational measure $\mu_{\rho}$) to define the spaces $\mathcal{U}^p_{\rho}$ of upper integrable functions and investigate their functional-analytic properties. The main contributions are: \begin{itemize} \item the precise construction of the $\rho$-upper-integrability spaces $\mathcal{U}^p_{\rho}(A;X)$ (and their Fr\'echet analogues), together with the natural semi-norms $\|\cdot\|_{\mathcal{U}^p_{\rho}}$; \item measure-style inequalities adapted to the variational measure $\mu_{\rho}$ (monotone continuity for ascending sets, Fatou-type lemma, and Chebyshev inequality) within the $\rho$-upper-integral framework; \item functional-analytic results: sequential completeness of $\mathcal{U}^p_{\rho}([a,b];X)$ when $X$ is sequentially complete (semi-normed case), and sequential completeness of $\mathcal{U}^p([a,b];X)$ when $X$ is a sequentially complete Fr\'echet space; and \item the closedness of the absolutely integrable subspace $L^p_{\rho}([a,b];X)$ inside $\mathcal{U}^p_{\rho}([a,b];X)$ (hence $L^p([a,b];X)$ is a closed Fr\'echet subspace of $\mathcal{U}^p([a,b];X)$ under the usual hypotheses). \end{itemize}