This question is about the distribution of the Conditional Expectation and how to use it to prove Second-Order Stochastic Dominance.
Let $(\Omega, \mathcal{F},P)$ be a probability space. For any $X \in L^{\infty}(\Omega, P)$, let $F_{X}$ be its cumulative distribution function and $q_{X}(t)=\inf \{x \in \mathbb{R}:F_{X}(x)>t\}$ its upper quantile function.
Consider a sub $\sigma-$field $\mathcal{G} \subset \mathcal{F}$ and let $Y:=E[X|\mathcal{G}]$ be a version of the conditional expectation of $X$ over $\mathcal{G}$.
I want to establish the second-order stochastic dominance of $Y$ over $X$. In order to do so, I could either show that
$\int_{-\infty}^{x}F_{Y}(t)dt \le \int_{-\infty}^{x}F_{X}(t)dt$ (with strict inequality at some $x \in \mathbb{R}$) or
$\int_{0}^{\lambda}q_{Y}(t)dt \ge \int_{0}^{\lambda}q_{X}(t)dt$ (with strict inequality at some $\lambda \in (0,1]$ - I guess)
Could someone give me hints on how to establish the above inequalities?