On Beck's Coloring for Measurable Functions

We study Beck-like coloring of measurable functions on a measure space Omega taking values in a measurable semigroup Delta. To any measure space Omega and any measurable semigroup Delta, we assign a graph (called a zero-divisor graph) whose vertices are labeled by the classes of measurable functions defined on Omega and having values in Delta, with two vertices f and g adjacent if f . g = 0 a.e.. We show that, if Omega is atomic, then not only the Beck's conjecture holds but also the domination number coincides to the clique number and chromatic number as well. We also determine some other graph properties of such a graph.