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
(iii)	$\mu^*_F$ is a D1858: Stieltjes outer measure with respect to $F$

The Stieltjes sigma-algebra on $\mathbb{R}$ with respect to $F$ is the D11: Set \begin{equation} \left\{ E \subseteq \mathbb{R} \mid \forall \, H \subseteq \mathbb{R} : \mu^*_F(E \cap H) + \mu^*_F(E^{\complement} \cap H) = \mu^*_F(H) \right\} \end{equation}