Let $F: \ol{\real} \to \ol{\real}$ be a *Stieltjes* function, that is: - $F$ is non-decreasing. - $F$ is *right*-continuous. - $F$ is finite on $\real$. then the mapping $ \mu_F\paren{\bigcup_{k = 1}^{n}(a_k, b_k]} = \sum_{k = 1}^{n}F(b_k) - F(a_k) $ where the intervals are disjoint, is a pre-measure on FDUs of h-intervals. *Proof (countable additivity)*. Let $ L = \bigcup_{i \in 1}^{n}(a_i, b_i] = \bigcup_{j \in \nat}(c_j, d_j] $ be unions of disjoint h-intervals. Let $m \in \nat$, then $ \bigcup_{i \in [n]}(a_i, b_i] \supset \bigcup_{j \le m}(c_j, d_j] $ Therefore $ \mu_F\paren{L} \ge \sum_{j \in \nat}\mu_F(c_j, d_j] $