Definition 16.3.1 (Integrable). Let $(X, \cm, \mu)$ be a measure space and $f: X \to \complex$ be a $(\cm, \cb_{\complex})$-measurable function, then $f$ is integrable if
\[\int \abs{f}d\mu < \infty\]
The set
\[\mathcal{L}^{1}(X, \cm, \mu; \complex) = \mathcal{L}^{1}(X, \cm, \mu) = \mathcal{L}^{1}(X; \complex) = \mathcal{L}^{1}(X) = \mathcal{L}^{1}(\mu; \complex) = \mathcal{L}^{1}(\mu)\]
is the vector space of $\mu$-integrable functions on $X$.
Proof. Let $f, g \in \mathcal{L}^{1}(X)$ and $\lambda \in \complex$, then
\[\int \abs{\lambda f + g}d\mu \le \int \abs{\lambda}\abs{f}+ \abs{g}d\mu = \lambda \int \abs{f}d\mu + \int \abs{g}d\mu\]
by Proposition 16.2.7.$\square$