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

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $\mathcal{C}$ be a nonempty family of square-integrable random variables on it. Write $\mathcal{S}(\mathcal{C})$, the closed mean-square span of $\mathcal{C}$, for the set of all square-integrable random variables $Q$ for which there are finite linear combinations $S_n$ of members of $\mathcal{C}$ with $\lVert S_n-Q\rVert_{2}\to0$, with $\lVert\cdot\rVert_{2}$ the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product.

1. (Closure) $\mathcal{S}(\mathcal{C})$ contains every member of $\mathcal{C}$, and is closed under finite linear combinations and under mean-square limits. Consequently, if every member of a nonempty family $\mathcal{C}'$ of square-integrable random variables lies in $\mathcal{S}(\mathcal{C})$, then $\mathcal{S}(\mathcal{C}')\subseteq\mathcal{S}(\mathcal{C})$.

2. (Measurability) Let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. Finite linear combinations of $\mathcal{G}$-measurable square-integrable random variables are $\mathcal{G}$-measurable; and if every member of $\mathcal{C}$ is $\mathcal{G}$-measurable, then every $Q\in\mathcal{S}(\mathcal{C})$ is almost surely equal to a $\mathcal{G}$-measurable square-integrable random variable.