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Definition D1858
Stieltjes outer measure

Let $F : \mathbb{R} \to \mathbb{R}$ be a D4364: Real function such that
 (i) $F$ is a D3249: Right-continuous function (ii) $F$ is an D5321: Standard-isotone basic real function
Let $\mathcal{I} = \{ (a, b] : a, b \in \mathbb{R}; a < b \}$ be the D3004: Set of right-closed real intervals in $\mathbb{R}$.
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 $$\mathcal{P}(\mathbb{R}) \to [0, \infty], \quad E \mapsto \inf_{C \in \mathcal{C}(E)} \sum_{(a, b] \in C} F(b) - F(a)$$

Let $F : \mathbb{R} \to \mathbb{R}$ be a D4364: Real function such that
 (i) $F$ is a D3249: Right-continuous function (ii) $F$ is an D5321: Standard-isotone basic real function
The Stieltjes outer measure on $\mathbb{R}$ with respect to $F$ is the D4361: Unsigned basic function $$\mathcal{P}(\mathbb{R}) \to [0, \infty], \quad E \mapsto \inf_{(a_0, b_0], (a_1, b_1], (a_2, b_2], \ldots \text{ is a cover for } E} \sum_{n = 0}^{\infty} F(b_n) - F(a_n)$$