The weak closure of the equimeasurable rearrangements of a measurable function

Abstract

If ( X, Λ, μ) is a finite measure space and f is in L 1 ( X, μ), then the σ( L 1, L ∞)-closure of the set Δ( f) of all measurable functions equimeasurable with f is shown to be the set to which g belongs if and only if there is a function equimeasurable with f which majorizes g (in the sense of the Hardy-Littlewood-Polya preorder relation) on the non-atomic part of X and which equals g on the union of the atoms of X. If ϱ is a saturated Fatou Banach function norm and Lϱ( X, μ) is universally rearrangement invariant such that L ∞ ⊂ Lϱ ⊂ L 1, then for all f in Lϱ the σ( Lϱ, Lϱ′)-closure of Δ ( f) is shown to be the same as the σ( L 1, L ∞)-closure of Δ ( f).