arXiv2025
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 $ρ$-variational measure $μ_ρ$) to define the spaces $\mathcal{U}^p_ρ$ of upper integrable functions and investigate their functional-analytic properties. The main contributions are: \begin{itemize} \item the precise construction of the $ρ$-upper-integrability spaces $\mathcal{U}^p_ρ(A;X)$ (and their Fréchet analogues), together with the natural semi-norms $\|\cdot\|_{\mathcal{U}^p_ρ}$; \item measure-style inequalities adapted to the variational measure $μ_ρ$ (monotone continuity for ascending sets, Fatou-type lemma, and Chebyshev inequality) within the $ρ$-upper-integral framework; \item functional-analytic results: sequential completeness of $\mathcal{U}^p_ρ([a,b];X)$ when $X$ is sequentially complete (semi-normed case), and sequential completeness of $\mathcal{U}^p([a,b];X)$ when $X$ is a sequentially complete Fréchet space; and \item the closedness of the absolutely integrable subspace $L^p_ρ([a,b];X)$ inside $\mathcal{U}^p_ρ([a,b];X)$ (hence $L^p([a,b];X)$ is a closed Fréchet subspace of $\mathcal{U}^p([a,b];X)$ under the usual hypotheses). \end{itemize}