ON REARRANGEMENT-INVARIANT AND MAJORANT HULLS OF AVERAGES OF REARRANGEMENT-INVARIANT AND MAJORANT IDEALS

Let F be a σ-algebra on [0, 1] generated by a countable partition of [0, 1], and let (·|F) denote the corresponding conditional expectation operator. For a subset Z of L1[0, 1] we denote by MZ (resp. NZ) the smallest interpolation (resp. rearrangement invariant) subspace of L1 containing Z. In Theorem 2.7 we show that for each f ε{lunate} L1 there exists a function g ε{lunate} L1 such that ME(Mj|F=Mg). While a similar result for Nf is not true in general (that is, NE(Nj|F is not always a principal ideal), Theorem 2.9, our main result, describes when it is the case. © 1992.