Special subsets of the real line
Let \(X\) be a subset of the reals, \(B(X)\) be the set of all Borel subsets of \(X\) and \(\mathcal A\) be a \(\sigma\)-algebra of subsets of \(X\). A Maharam submeasure \(\nu\) on \(\mathcal A\) is a mapping from \(\mathcal A\) into [0,1] such that: 1) \(\nu(\emptyset)=0\); 2) \(\nu(\{x\})=0\), \(\forall x\in X\); 3) if \(E\) and \(F\) belong to \(\mathcal A\) and \(E\subset F\), then \(\nu(E)\leq\nu(F)\); 4) if \(E,F\in\mathcal A\), then \(\nu(E\cup F)\leq\nu(E)+\nu(F)\) and 5) if \((E_ n)\) is a sequence of sets of \(\mathcal A\) and \(\bigcap_{k\in\omega}\bigcup_{n\geq k}(E_ n\Delta E)=\emptyset\), then \(\lim_{n\to\infty}\nu(E_ n)=\nu(E)\). The author deals with topological solutions of the following control problem: for a given \(\sigma\)-algebra \(\mathcal A\) and a Maharam submeasure \(\nu\) on \(\mathcal A\), does there exist a probability measure \(\mu\) on \(\mathcal A\) such that \(\nu(A)=0\) iff \(\mu(A)=0\), for all \(A\in\mathcal A\)? Solutions depend on Martins axiom and the negation of continuum hypothesis.
scientific article; zbMATH DE number 11456