Density topologies for strictly positive Borel measures (Q2665200)

Any abstract density topology is generated by a lower density operator \(\Phi\) on the space \((X, \mathcal{S}, \mathcal{I})\), where \( \mathcal{S}\) is a \(\sigma\)-algebra of subsets of \(X\) and \( \mathcal{I}\) is \(\sigma\)-ideal included in \( \mathcal{S}\). A basic example of such a topology is the classical density topology \( \mathcal{T}_d\) connected with the Lebesgue measure. A generalization of the density with respect to the Lebesgue measure can be obtained for other measures. For any complete measure \(\mu\) denote by \(S_{\mu}\) the \(\sigma\)-algebra of all \(\mu\)-measurable sets and by \(\mathcal{N}_{\mu}\) the \(\sigma\)-ideal of all \(\mu\)-null sets. Let \(\mu\) be any Borel measure which is finite, complete and regular. For such a measure there are defined: the \(\mu\)-density \(d_{\mu}\), the operator \(\Phi_{\mu}\) and the family \(\mathcal{T}_{\mu}=\{A\in S_{\mu}\colon A\subset \Phi_{\mu}(A)\}\). If \(\mu\) is a strictly positive Borel measure then \(\Phi_{\mu}\) is a lower \(\mu\)-density operator on \(([0,1], S_{\mu}, \mathcal{N}_{\mu})\) and the family \(\mathcal{T}_{\mu}\) is a topology on \([0,1]\). Moreover, \(([0,1], \mathcal{T}_{\mu} )\) is regular but not normal. In some cases it is Tychonoff. If \(\nu\) is any atomless measure then the topology \(\mathcal{T}_{\nu}\) is homeomorphic to \( \mathcal{T}_d\). However, there are measures \(\mu\) generating topologies quite different from \( \mathcal{T}_d\) (for instance: separable and not connected, not Lindelöf but separable).