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In a recent paper, M. Valdivia has shown that if \(\Omega \) is a compact
\(k\)-dimensional interval in \(\mathbb{R}^{k}\), \(\mathcal{A}\) is the algebra of
Jordan measurable subsets of \(\Omega \),\
\(\mathcal{A}=\cup _{n}\mathcal{A}_{n}\), with
\(\mathcal{A}_{n}\subset \mathcal{A}_{n+1}\),\
\(n\in \mathbb{N}\), then there exists
\(m\in \mathbb{N}\) such that if \(H\) is a set of bounded additive complex measures
defined in \(\mathcal{A}\) such that, for each \(A\in \mathcal{A}_{m}\),\
\(\sup \{\left\vert\lambda (A)\right\vert :\lambda \in H\}<\infty \), then
\(\sup \{\left\vert
\lambda \right\vert (\Omega ):\lambda \in H\}<\infty \), where \(\left\vert
\lambda \right\vert \) is the variation of \(\lambda \). M. Valdivia says that
``The proof of this theorem can be extended to more general situations''.
We will discuss some extension of this results. In particular we will prove
that if \(\mathcal{A}\) is a \(\sigma \)-algebra defined on a set \(\Omega \) and
if
- \(\mathcal{A}=\bigcup _{n_{1}}\mathcal{A}_{n_{1}}\), with \(\mathcal{A}
_{n_{1}}\subset \mathcal{A}_{n_{1}+1}\), \(n_{1}\in
\mathbb{N}\),
- each \(\mathcal{A}_{n_{1}}=\cup _{n_{2}}\mathcal{A}_{n_{1},n_{2}}\), with \(
\mathcal{A}_{n_{1},n_{2}}\subset \mathcal{A}_{n_{1},n_{2}+1}\), \(
(n_{1},n_{2})\in
\mathbb{N}^{2}\),
- ....,
- and each \(\mathcal{A}_{n_{1},n_{2},\cdots ,n_{p-1}}=\bigcup _{n_{p}}\mathcal{A
}_{n_{1},n_{2},\cdots ,n_{p-1}n_{p}}\), with \(\mathcal{A}_{n_{1},n_{2},\cdots
,n_{p-1}n_{p}}\subset \mathcal{A}_{n_{1},n_{2},\cdots ,n_{p-1}n_{p}+1}\), \(
(n_{1},n_{2},\cdots ,n_{p})\in
\mathbb{N}^{p}\),
then there exists \((m_{1},m_{2},\cdots ,m_{p})\in \mathbb{N}^{p}\) such that if \(H\) is a set of bounded additive measures defined in \(\mathcal{A}\) such that for each \(A\in \mathcal{A}_{m_{1},m_{2},\cdots
,m_{p-1}m_{p}}\), \(\sup \{\left\vert \lambda (A)\right\vert :\lambda \in
H\}<\infty \), then \(\sup \{\left\vert \lambda \right\vert (\Omega ):\lambda
\in H\}<\infty \), where \(\left\vert \lambda \right\vert \) is the variation
of \(\lambda \). Additionally, if we continue this decomposition process then
there exists a sequence \((m_{q})_{q}\) such for each \(p\in \mathbb{N}\) the finite sequence \((m_{1},m_{2},\cdots ,m_{p})\) has the previous
boundedness property.
Some applications of this boundedness result will be presented.
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