Some applications of the adjoint to lattice regular measures (Q1059728)

Let X be an abstract set and \({\mathcal L}^ a \)lattice of subsets of X. M(\({\mathcal L})\) denotes all bounded finitely additive measures on \({\mathcal A}({\mathcal L})\), the algebra generated by \({\mathcal L}\); MR(\({\mathcal L})\), denotes the \({\mathcal L}\)-regular elements of M(\({\mathcal L})\), and MR(\(\sigma\),\({\mathcal L})\), those elements of MR(\({\mathcal L})\) that are countably additive. \(C_ b({\mathcal L})\) denotes the bounded \({\mathcal L}\)- continuous functions on X. Next, let X and Y be two abstract sets with lattices \({\mathcal L}_ 1\) and \({\mathcal L}_ 2\) respectively, and \(T:X\to Y\) an \({\mathcal L}_ 1\)-\({\mathcal L}_ 2\) continuous map. Then \(\hat T:MR({\mathcal L}_ 1)\to MR({\mathcal L}_ 2),\) by \(\hat T\mu =\mu T^{-1}.\) If \({\mathcal L}_ 1\) and \({\mathcal L}_ 2\) are also both delta-normal lattices, then T induces a linear mapping \(A:C_ b({\mathcal L}_ 2)\to C_ b({\mathcal L}_ 1),\) and the Alexandroff representation theorem gives \(A':MR({\mathcal L}_ 1)\to MR({\mathcal L}_ 2)\) where A' is the adjoint map. One always has that \(A'\mu \geq \hat T\mu\) on \({\mathcal L}_ 2\), and if T is subjective and \(T^{-1}({\mathcal L}_ 2)\) semiseparates \({\mathcal L}_ 1\) then \(A'=T.\) Properties of A' and T are investigated in the paper. Conditions are given under which A' and T map particular subsets of MR(\({\mathcal L}_ 1)\) into particular subsets of MR(\({\mathcal L}_ 2):\) for example MR(\(\sigma\),\({\mathcal L}_ 1)\) into MR(\(\sigma\),\({\mathcal L}_ 2)\) and MR(\({\mathcal L}_ 1)-MR(\sigma,{\mathcal L}_ 1)\) into MR(\({\mathcal L}_ 2)- MR(\sigma,{\mathcal L}_ 2).\) Specific applications are given to the case where X and Y are topological spaces, and \({\mathcal L}_ 1\), \({\mathcal L}_ 2\) particular lattices related to the topologies.