The main theorem is the following: Let X be a perfect complete separable metric space and let \(T_ 0\) be the \(\sigma\)-ideal of subsets of X such that (1) there exists a compact \(X_ 0\subseteq X\) such that \(X_ 0\not\in T_ 0\), and (2) for each countable set \(A\subseteq X\) there is a \(G_{\delta}\)-set B such that \(A\subseteq B\in T_ 0\). Then each \(\sigma\)-ideal T such that \(T\subseteq T_ 0\) has order \(\omega_ 1.\)
The theorem of Mauldin is the special case in which \(X=[0,1]\) and \(T=T_ 0\) is the ideal of sets of Lebesgue measure zero. The proof uses the method of Mauldin, but is not given here. A number of remarks, corollaries and problems are also included.