Conditional Mean-Square Optimality Restricted to an Event of the Conditioning Sigma-Algebra

Adopt the setting and notation of the conditional mean-square optimality lemma: a probability space $(\Omega,\mathcal{F},P)$, a sub-$\sigma$-algebra $\mathcal{G}$ of $\mathcal{F}$, a natural number $k\ge1$, a tuple $X=(X^1,\dots,X^k)$ of square-integrable random variables, a fixed tuple $M=(M^1,\dots,M^k)$ of conditional expectations $M^\gamma$ of $X^\gamma$ given $\mathcal{G}$, the error tuple $\varepsilon=X-M$, a symmetric positive semidefinite real $k\times k$ matrix $R$ with entries $R_{\gamma\delta}$, the pointwise quadratic form $U\cdot(RV)=\sum_{\gamma,\delta=1}^{k}R_{\gamma\delta}U^\gamma V^\delta$ for tuples $U,V$ of square-integrable random variables, and an admissible tuple $Y=(Y^1,\dots,Y^k)$: each $Y^\gamma$ is square-integrable and almost surely equal to a $\mathcal{G}$-measurable square-integrable random variable, with $\mathcal{G}$-measurability as in the existence and uniqueness theorem for conditional expectation. Let $G\in\mathcal{G}$, write $\mathbf{1}_{G}$ for the function equal to $1$ on $G$ and $0$ off $G$, and write $\mathbb{E}$ for the expectation. Then all the products appearing below are integrable and:

1. (Restricted orthogonal decomposition.)

$$\mathbb{E}\big[\mathbf{1}_{G}\,(Y-X)\cdot\big(R\,(Y-X)\big)\big]=\mathbb{E}\big[\mathbf{1}_{G}\,\varepsilon\cdot(R\,\varepsilon)\big]+\mathbb{E}\big[\mathbf{1}_{G}\,(Y-M)\cdot\big(R\,(Y-M)\big)\big].$$

2. (Restricted lower bound.)

$$\mathbb{E}\big[\mathbf{1}_{G}\,(Y-X)\cdot\big(R\,(Y-X)\big)\big]\ \ge\ \mathbb{E}\big[\mathbf{1}_{G}\,\varepsilon\cdot(R\,\varepsilon)\big].$$

3. (Index expansion and independence of the choice.)

$$\mathbb{E}\big[\mathbf{1}_{G}\,\varepsilon\cdot(R\,\varepsilon)\big]=\sum_{\gamma=1}^{k}\sum_{\delta=1}^{k}R_{\gamma\delta}\,\mathbb{E}\big[\mathbf{1}_{G}\,\varepsilon^\gamma\varepsilon^\delta\big],$$

and each $\mathbb{E}[\mathbf{1}_{G}\varepsilon^\gamma\varepsilon^\delta]$ is unchanged if the conditional expectations $M^\gamma$ are replaced by any other conditional expectations of the $X^\gamma$ given $\mathcal{G}$.