On rearrangement invariant and majorant hulls of averages of rearrangement invariant and majorant ideals (Q1206988)

Let \({\mathcal F}\) be a \(\sigma\)-algebra on \([0,1]\) generated by a countable partition of \([0,1]\), and let \(E(\cdot\mid {\mathcal F})\) denote the corresponding conditional expectation operator. For a subset \(Z\) of \(L^ 1 [0,1]\) the author denotes by \(M_ Z\) (resp. \(N_ Z\)) the smallest interpolation (resp. rearrangement invariant) subspace of \(L^ 1\) containing \(Z\). He shows that for each \(f\in L_ 1\) there exists a function \(g\in L^ 1\) such that \(M_{E(M_ f\mid {\mathcal F})} =M_ g\). While a similar result for \(N_ f\) is not true in general (that is, \(N_{E(N_ f\mid {\mathcal F})}\) is not always a principal ideal), his main result describes when it is the case.