Definition (Integrable)

Let $f:X\to \mathbb{R}$ measurable. $f$ is called integrable if $$\int\limits _{X}|f| \, d\mu<\infty$$

Definition (Set of all Integrable Functions)

Let $(X,\mathcal{F},\mu)$ be a measure space. We define $$\mathscr{L}^{1}(X,\mathcal{F},\mu)$$to be the set of all integrable functions $f:X\to \mathbb{R}$.