Since $\mathcal{S} \subseteq \mathcal{F}$, then it is left to be shown that $\mathcal{S}$ is a D84: Sigma-algebra on $X$. For this, we confirm that $\mathcal{S}$ satisfies the properties required of a sigma-algebra on $X$. These properties are established in the results

(i)	R4466: Whole space is a stationary measurable set
(ii)	R4467: Empty set is a stationary measurable set
(iii)	R4437: Complement of stationary measurable set is stationary
(iv)	R4469: Countable union of stationary measurable sets is stationary

$\square$