  We consider modified scalar curvature functions for Riemannian manifolds
equipped with smooth measures. Given a Riemannian submersion whose fiber
transport is measure-preserving up to constants, we show that the modified
scalar curvature of the base is bounded below in terms of the scalar curvatures
of the total space and fibers. We give an application concerning the scalar
curvature of a smooth limit space arising in a bounded curvature collapse.
