Density Properties of Finitely Additive Measures

We consider the space of all finitely additive measures of bounded variation. The family of all measurable sets is an algebra. We study two cases. The first case corresponds to a non-atomic non-negative measure $\lambda$. The second case corresponds to a finite range  non-negative measure $\lambda$. It is shown for the first case that the set of all step functions (and the set of all uniform limits of ones) such that $\lambda$-integrals equal to the unit allows an embedding into the compact set in the form of an everywhere dense subset. This compact set is the unit ball of weakly absolutely continuous measures with respect to $\lambda$. In this case the ball equals the weak-star closure of the unit sphere of such measures. For the second case it is shown that the set of all step functions (and the set of all uniform limits of ones) such that $\lambda$-integrals equal to the unit allows an embedding into the compact set in the form of an everywhere dense subset. This compact set is the unit sphere of weakly absolutely continuous measures with respect to $\lambda$. In this case the unit sphere is closed with respect to weak-star topology.