On Beck's Coloring for Measurable Functions

We study Beck-like coloring of measurable functions on a measure space Ω taking values in a measurable semigroup ∆. To any measure space Ω and any measurable semigroup ∆, we assign a graph (called a zero-divisor graph) whose vertices are labeled by the classes of measurable functions defined on Ω and having values in ∆, with two vertices f and g adjacent if f · g = 0 a.e.. We show that, if Ω 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.