Theorem

The Set of all Integrable Functions, $\mathscr{L}^{1}(X,\mathcal{F},\mu)$ is a real-vector space. Furthermore the map $$\begin{align*} \mathscr{L}^{1}(X,\mathcal{F},\mu)&\to \mathbb{R}\\ f&\mapsto \int\limits f \, d\mu \end{align*}$$ is $\mathbb{R}$-linear.