(i) | $F$ is a D3249: Right-continuous function |
(ii) | $F$ is an D5321: Standard-isotone basic real function |
Let $\mathcal{C}(E) \subseteq \mathcal{P}(\mathcal{I})$ denote the D3041: Set of countable set covers of any $E \subseteq \mathbb{R}$ with respect to $\mathcal{I}$.
The Stieltjes outer measure on $\mathbb{R}$ with respect to $F$ is the D4361: Unsigned basic function \begin{equation} \mathcal{P}(\mathbb{R}) \to [0, \infty], \quad E \mapsto \inf_{C \in \mathcal{C}(E)} \sum_{(a, b] \in C} F(b) - F(a) \end{equation}