The set of all maximal ideals of the ring $mathcal{M}(X,mathcal{A})$ of real valued measurable functions on a measurable space $(X,mathcal{A})$ equipped with the hull-kernel topology is shown to be homeomorphic to the set $hat{X}$ of all ultrafilters of measurable sets on $X$ with the Stone-topology. This yields a complete description of the maximal ideals of $mathcal{M}(X,mathcal{A})$ in terms of the points of $hat{X}$. It is further shown that the structure spaces of all the intermediate subrings of $mathcal{M}(X,mathcal{A})$ containing the bounded measurable functions are one and the same and are compact Hausdorff zero-dimensional spaces. It is observed that when $X$ is a $P$-space, then $C(X) = mathcal{M}(X,mathcal{A})$ where $mathcal{A}$ is the $sigma$-algebra consisting of the zero-sets of $X$.
تحميل البحث