Proof of The Closed Mean-Square Span of a Family of Random Variables

Throughout, the triangle inequality is from Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm, and mean-square limits are unique up to almost-sure equality: if $\lVert S_n-Q\rVert_2\to0$ and $\lVert S_n-Q'\rVert_2\to0$ then $\lVert Q-Q'\rVert_2=0$, so $Q=Q'$ almost surely by the null-equivalence of Square-Integrable Random Variables and the Mean-Square Inner Product.

Claim 1. Members of $\mathcal{C}$ lie in $\mathcal{S}(\mathcal{C})$ (constant approximating sequences). Linear combinations: if $\lVert S_n-Q\rVert_2\to0$ and $\lVert S'_n-Q'\rVert_2\to0$ with $S_n,S'_n$ finite linear combinations of members of $\mathcal{C}$, then for reals $a,a'$, $aS_n+a'S'_n$ is again such a combination and $\lVert(aS_n+a'S'_n)-(aQ+a'Q')\rVert_2\le|a|\lVert S_n-Q\rVert_2+|a'|\lVert S'_n-Q'\rVert_2\to0$; induction extends this to any finite combination. Limits: if $Q_n\in\mathcal{S}(\mathcal{C})$ and $\lVert Q_n-Q\rVert_2\to0$ with $Q$ square-integrable, choose for each $n$ a finite linear combination $S_n$ of members of $\mathcal{C}$ with $\lVert Q_n-S_n\rVert_2\le1/n$; then $\lVert S_n-Q\rVert_2\le1/n+\lVert Q_n-Q\rVert_2\to0$, so $Q\in\mathcal{S}(\mathcal{C})$. Consequence: if every member of $\mathcal{C}'$ lies in $\mathcal{S}(\mathcal{C})$, then every finite linear combination of members of $\mathcal{C}'$ lies in $\mathcal{S}(\mathcal{C})$ (closure under combinations), and hence so does every mean-square limit of such combinations (closure under limits): $\mathcal{S}(\mathcal{C}')\subseteq\mathcal{S}(\mathcal{C})$.

Claim 2. Combinations: it suffices that $cX$ and $X+Y$ are $\mathcal{G}$-measurable for $\mathcal{G}$-measurable $X,Y$ and real $c$. For $c>0$, the set where $cX>\lambda$ is the set where $X>\lambda/c$, which lies in $\mathcal{G}$; for $c<0$ it is the set where $X<\lambda/c$; for $c=0$ it is $\Omega$ or the empty set. The set where $X+Y>\lambda$ is the countable union over rational $q$ of the intersections of the sets where $X>q$ and where $Y>\lambda-q$, which lies in $\mathcal{G}$. By the generator criterion applied on the measurable space $(\Omega,\mathcal{G})$, these level-set memberships give $\mathcal{G}$-measurability.

Limits: let $Q\in\mathcal{S}(\mathcal{C})$ with approximating combinations $S_n$, each $\mathcal{G}$-measurable by the previous paragraph. The sequence $(S_n)$ is Cauchy in mean square (triangle inequality through $Q$), so Mean-Square Completeness of Square-Integrable Random Variables (Riesz-Fischer), applied with the sub-$\sigma$-algebra $\mathcal{G}$, yields a $\mathcal{G}$-measurable square-integrable $Q'$ with $\lVert S_n-Q'\rVert_2\to0$; by uniqueness of mean-square limits, $Q=Q'$ almost surely. $\blacksquare$