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

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 ${\mathcal{U}}_{\rho }^p$ 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 ${\mathcal{U}}_{\rho }^p(A;X)$ (and their Fréchet analogues), together with the natural semi-norms $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{{\mathcal{U}}_{\rho }^p}$; \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 ${\mathcal{U}}_{\rho }^p([a,b];X)$ 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 $L_{\rho }^p([a,b];X)$ inside ${\mathcal{U}}_{\rho }^p([a,b];X)$ (hence $L^p([a,b];X)$ is a closed Fréchet subspace of ${\mathcal{U}}^p([a,b];X)$ under the usual hypotheses). \end{itemize}