Ideals in rings and intermediate rings of measurable functions

Abstract

The set of all maximal ideals of the ring [Formula: see text] of real valued measurable functions on a measurable space [Formula: see text] equipped with the hull-kernel topology is shown to be homeomorphic to the set [Formula: see text] of all ultrafilters of measurable sets on [Formula: see text] with the Stone-topology. This yields a complete description of the maximal ideals of [Formula: see text] in terms of the points of [Formula: see text]. It is further shown that the structure spaces of all the intermediate subrings of [Formula: see text] containing the bounded measurable functions are one and the same and are compact Hausdorff zero-dimensional spaces. It is observed that when [Formula: see text] is a [Formula: see text]-space, then [Formula: see text], where [Formula: see text] is the [Formula: see text]-algebra consisting of the zero-sets of [Formula: see text].