Lebesgue Space is Vector Space

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $\mathcal L^p \left({\mu}\right)$ be Lebesgue $p$-space for $\mu$.

Then $\mathcal L^p \left({\mu}\right)$ is a vector subspace of $\mathcal M \left({\Sigma}\right)$, the space of $\Sigma$-measurable functions.

In particular, it is a vector space.