Suppose we have real-valued random variables \(X\) and \(Y\) where \(\mathbb{E}[X]\) and \(\mathbb{E}[Y]\) are finite (i.e. \(X\) and \(Y\) are integrable). Then for any values our r.v.s \(X\) and \(Y\) can take on we have \(\vert X + Y \vert \leq \vert X \vert + \vert Y \vert \implies \mathbb{E}[\vert X + Y \vert] \leq \mathbb{E}[\vert X \vert + \vert Y \vert ] = \mathbb{E}[\vert X \vert] + \mathbb{E}[\vert Y \vert]\).
This is not a terribly interesting inequality, so we’ll derive another expectation inequality relating differences. First note that for any r.v. \(W\) we have \(\vert \mathbb{E}[W] \vert \leq \mathbb{E}[\vert W \vert]\) as:
\[\vert \mathbb{E}[W] \vert = \vert \int_{-\infty}^{\infty} w f_{W}(w) dw \vert \leq \int_{-\infty}^{\infty} \vert w \vert f_{W}(w) dw = \mathbb{E}[\vert W \vert]\]An alternative explanation is that this is an example of Jensen’s inequality as the absolute value function is convex. Now defining r.v. \(W = X - Y\), we have:
\[\vert \mathbb{E}[X] - \mathbb{E}[Y] \vert = \vert \mathbb{E}[W] \vert \leq \mathbb{E}[\vert W \vert] = \mathbb{E}[\vert X - Y \vert]\]or \(\vert \mathbb{E}[X] - \mathbb{E}[Y] \vert \leq \mathbb{E}[\vert X - Y \vert]\).
I found this particularly useful when trying to understand convergence in expectation (e.x. this limit on page 6).