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 \begin{equation} \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) \end{equation}