Measurable (α,β)-spaces with vanishing S-curvature

In this paper, we give a necessary and sufficient condition that an (α,β)-space admits a measure μ with vanishing S-curvature everywhere. It is shown that the measure of such an (α,β)-space must coincide with the Riemannian volume measure of α up to a constant multiplication. Moreover, the characterization of (α,β)-spaces admitting a measure μ with weak isotropic S-curvature is given.