Slightly regular measures and measureable sets (Q670542)

The setting is on a lattice \(\ell\) of subsets of \(X\ne \emptyset\) with both \(X\) and \(\emptyset\) in \(\ell\), with a measure \(\mu\) defined on the algebra \(\mathcal{A}(\ell) \) generated by \(\ell\). On the set \(M(\ell)\) of non-trivial finitely additive measures on \(\mathcal{A}(\ell)\), a measure \(\mu\) is \textit{slightly regular} if whenever \( \cap_i L_i = L\), \(L\) and \(L_i \in \ell\), then \( \mu(L) = \inf \mu(L_i)\). This densely written paper studies the slightly regular measure \(M_S(\ell) \subset M(\ell)\) in relation to other more studied measures. For example, on the set of \(\ell\)-regular measures \(M_R(\ell)\) (i.e., \(\mu (E)= \sup\{\mu(L)|L \subset E, L \in \ell\}\)), the \(\sigma\)-smooth measures \(M_\sigma (\ell)\), (i.e., for any sequence \(\{L_n\}, L_n \in \ell, L_n \downarrow \emptyset, \lim_{n}\mu(L_n) = 0\)), and the \(\sigma-\)smooth \(\ell-\)regular measures \(M_R^\sigma(\ell)=M_R(\ell) \cap M_\sigma(\ell)\), we have \(M_R^\sigma(\ell) \subset M_S(\ell)\subset M_\sigma(\ell)\). The author found conditions under which two slightly regular measures are equal, and also conditions that a slightly regular measure is \(\sigma\)-smooth and \(\ell\)-regular. For \(\mu \in M_\sigma(\ell)\), various outer and inner measures are induced, each with its associated measurable sets (for example, a set \(E \subseteqq X\) is \(\mu''\)-measurable if for any \(A\in X\), \(\mu''(A) = \mu''(A\cap E)) +\mu''(A \cap E')\), for all \(A\subset X\), \(\mu''\) being an induced outer-measure). When \(\ell\) is a \(\delta\)-lattice (\(\ell\) is closed under countable intersection), the paper investigates the relationships between the different induced inner and outer measures when \(\mu\) is slightly regular, and also when \(\mu \in M_S(\ell) \cap M_\sigma(\ell)\), where the measure is shown to be equal on \(\ell\) to one of its induced outer measure. With these results, conditions are found for a \(\sigma-\)smooth measure to be slightly regular.