Completeness of the space of absolutely and upper integrable functions with values in a semi-normed space
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909785383763968 |
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| author | Maza, Rodolfo E. |
| author_facet | Maza, Rodolfo E. |
| contents | 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} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05694 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Completeness of the space of absolutely and upper integrable functions with values in a semi-normed space Maza, Rodolfo E. Functional Analysis Primary 26A42 28B05 46G10, Secondary 46A04 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} |
| title | Completeness of the space of absolutely and upper integrable functions with values in a semi-normed space |
| topic | Functional Analysis Primary 26A42 28B05 46G10, Secondary 46A04 |
| url | https://arxiv.org/abs/2506.05694 |