In here, stated as a theorem is:
Let $\mathcal{G}\subseteq\mathcal{F}$ be $\sigma$-algebras, $X\in L_2$ be a random variable on the probability space. Then, there exists a random variable $Y\in\mathcal{G}$ such that
$E||X-Y||_2=\inf_{Z\in\mathcal{G}}E||X-Z||_2$
How do I prove such an element exists up to almost surely equivalence? Also, how can I show that this is $E[X|\mathcal{G}]$. In this appendix, $E[X|\mathcal{G}]$ is defined as this, but what if we start with the usual definition that this conditional expectation satisfies the properties that for all $G\in\mathcal{G}$, $\int_{G} E[X|\mathcal{G}]dP=\int_G XdP$ and $E[X|\mathcal{G}]$ is $\mathcal{G}$-measurable and integrable?