Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{G}$ be a sub- $\sigma$-algebra of $\mathcal{F}$. Show that, for any random variable $X \in L^{2}(\mathbb{P})$, there exists a $\mathcal{G}$-measurable random variable $Y \in L^{2}(\mathbb{P})$ such that $\mathbb{E}((X-Y) Z)=0$ for all $\mathcal{G}$-measurable random variables $Z \in L^{2}(\mathbb{P})$.

[You may assume without proof the completeness of $L^{2}(\mathbb{P}) .$ ]

Let $(G, X)$ be a Gaussian random variable in $\mathbb{R}^{2}$, with mean $(\mu, \nu)$ and covariance $\operatorname{matrix}\left(\begin{array}{cc}u & v \\ v & w\end{array}\right)$. Assume that $\mathcal{F}=\sigma(G, X)$ and $\mathcal{G}=\sigma(G)$. Find the random variable $Y$ explicitly in this case.