Let and be two simple graphs. Let (, 1) and (, 2) be two vertex measure spaces. In this paper we introduce a σ algebra 1 2 , which consists of all vertex induced sub graphs of , and it contains every vertex measurable rectangle graph of the form H 1 H 2 , H 1 ∈ 1 and H 2 ∈ 2. Here, we prove 1 2 is the smallest σ algebra of such that the maps and defined by and for all vertex measurable graphs H in and K in respectively are measurable.