I am reading Jacod and Protter's Probability Essentials and I am struggling to understand the following string of inequalities:
where $L^1$ is the space of real valued random variables on ($\Omega$, $A$, $P$) for $\sigma$-algebra $A$ and probability measure $P$ that have a finite expectation. Note that in this chapter, $A$ is defined s.t. $A$=$2^\Omega$ for a countable $\Omega$.
Specifically, I do not understand why the expecation would be "bifurcated" in to the sum of the two subsets of the image of $\omega$ such that $\lvert X( \omega) \rvert <1$ and $\lvert X( \omega) \rvert \ge 1$.
Edit: I am still struggling to see how I can bound $\sum_{\lvert X( \omega) \rvert <1} X(\omega)p_\omega$ or $\sum_{\lvert X( \omega) \rvert <1} \lvert X(\omega)p_\omega \rvert$ for that matter by $\sum_\omega p_\omega$ and $\sum_{\lvert X( \omega) \rvert \ge 1} \lvert X(\omega)p_\omega \rvert$ by $\sum_\omega (X(\omega))^2 p_\omega$. Apologies for not making this clearer in my original post.
Quirky. This should be starting like this: $$\sum_{\omega}|X(\omega)|p_{\omega} = \sum_{|X(\omega)| < 1}\color{red}{|}X(\omega)\color{red}{|}p_{\omega} + \sum_{|X(\omega)| \geq 1}\color{red}{|}X(\omega)\color{red}{|}p_{\omega} \leq \ldots$$ (with the rest as is). The splitting is deliberate - it just allows to upper-bound each sum as is done.
Namely, $$\sum_{|X(\omega)| < 1}|X(\omega)|p_{\omega} \leq \sum_{|X(\omega)| < 1}p_{\omega} \leq \sum_{\omega}p_{\omega}$$ and, as $|x| \leq x^2$ when $|x| \geq 1$, $$\sum_{|X(\omega)| \geq 1}|X(\omega)|p_{\omega} \leq \sum_{|X(\omega)| \geq 1}(X(\omega))^2 p_{\omega} \leq \sum_{\omega}(X(\omega))^2 p_{\omega}.$$