Upper and lower densities have the strong Darboux property

We show that every upper density μ⋆ has the strong Darboux property, and so does the associated lower density, where a function f:P(N)→R is said to have the strong Darboux property if, whenever X⊆Y⊆N and a∈[f(X),f(Y)], there is a set A such that X⊆A⊆Y and f(A)=a. In fact, we prove the above under the assumption that the monotonicity of μ⋆ is relaxed to the weaker condition that μ⋆(X)≤1 for every X⊆N.