The main result of the present paper, strengthening an earlier result of the authors [Arch. Math. 43, 373-376 (1984; Zbl 0603.31009)], essentially says that (under suitable assumptions) for each measure \(\mu\) in a given class \(M_ 0\) and for some reference measure \(\sigma\), the Radon- Nikodým derivative \(d\mu/d\sigma\) is equal to a limit along a directed set \(I\) of ratios \(R_ i(\mu,\sigma)\), \(i\in I\) (depending on \(\mu\) and \(\sigma)\) \(\sigma\)-almost everywhere. The authors obtain two conditions each of which being equivalent to the just mentioned property. They give interesting applications of their main result, including boundary limit theorems in potential theory, martingale limit theory and density theorems in measure theory.