Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $X: \Omega \to \mathbb{R}$ be a random variable. Write $X^+ = \max(X, 0)$ and $X^- = \max(-X, 0)$, so that $X = X^+ - X^-$ and $|X| = X^+ + X^-$. If $\min(\mathbb{E}[X^+], \mathbb{E}[X^-]) < \infty$, the expectation of $X$ is
\begin{align}
\mathbb{E}[X] = \mathbb{E}[X^+] - \mathbb{E}[X^-] \in [-\infty, \infty].
\end{align}
The random variable $X$ is called integrable if $\mathbb{E}[|X|] = \mathbb{E}[X^+] + \mathbb{E}[X^-] < \infty$, in which case $\mathbb{E}[X] \in \mathbb{R}$.