Let u be a δ-subharmonic function with associated measure µ, and let v be a superharmonic function with associated measure ν, on an open set E. For any closed ball B(x,r), of centre x and radius r, contained in E, let M(u, x, r) denote the mean value of u over the surface of the ball. We prove that the upper and lower limits as s, t → 0 with 0 <s<t of the quotient (M(u, x,s)−M(u, x,t))/(M(v,x,s)−M(v,x,t)), lie between the upper and lower limits as r → 0+ of the quotient µ(B(x,r))/ν(B(x,r)). This enables us to use some well-known measure-theoretic results to prove new variants and generalizations of several theorems about δ-subharmonic functions.