On maximal measures with respect to a lattice (Q804736)

Let L be a lattice of subsets of an abstract set X, A(L) the algebra generated by L, M(L) the finitely additive bounded measures on A(L), \(M_ R(L)\) the subsets of M(L), consisting of L-regular measures, \(M_{\sigma}(L)\) the set of \(\sigma\)-smooth measures on L of M(L), \(M_ R^{\sigma}(L)\)- the set of L-regular \(\sigma\)-smooth measures on A(L) of M(L). For \(m\in M(L)\) define \(\lambda (E)=\sup \{m(C)\); \(C\subset E\), \(C\in L\}\) and \(\hat m(E)=\inf \{\lambda (C');\quad E\subset C',\quad C\in L\}.\) The following results are proved: a) If L is a normal lattice, \(m\in M(L)\), \(n_ 1,n_ 2\in M_ R(L)\) and \(m(C)\leq n_ 1(C)\leq n_ 2(C)\) for all \(C\in L\) with \(m(X)=n_ 1(X)=n_ 2(X),\) then \(n_ 1=n_ 2;\) b) If L is normal, countably paracompact then \(m\in M_{\sigma}(L)\) implies \(\hat m\in M_ R^{\sigma}(L);\) c) If L is normal, \(m\in M_{\sigma}(L)\), then \(\hat m\) restricted to A(L) is in \(M_{\sigma}(L')\cap M_ R(L)\).