L1 contraction of conditional expectation ($\Vert \mathbb{E}[Y\mid \mathcal{G}]{\Vert }_1\le \Vert Y{\Vert }_1$)

For an integrable random variable $Y$, the conditional expectation satisfies $|\mathbb{E}[Y\mid \mathcal{G}]|\le \mathbb{E}[|Y|\mid \mathcal{G}]$ almost surely. Taking expected values gives the L1 contraction of conditional expectation. Applying it to $Y_n-Y$ shows that convergence in L1 is preserved by conditional expectation with respect to any fixed sigma-algebra.