All amenable factors can be constructed using the group-measure space construction of Murray and von Neumann for a single ergodic transformation. In fact they are precisely the factors arising as crossed products by free ergodic actions of "Z" or "Z/nZ" on abelian von Neumann algebras "L"∞("X"). Type I factors occur when the measure space "X" is atomic and the action transitive. When "X" is diffuse or non-atomic, it is equivalent to [0,1] as a measure space. Type II factors occur when "X" admits an equivalent finite (II1) or infinite (II∞) measure, invariant under an action of "Z". Type III factors occur in the remaining cases where there is no invariant measure, but only an invariant measure class: these factors are called Krieger factors.