Lebesgue Space is Vector Space

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\map {\LL^p} \mu$ be Lebesgue $p$-space for $\mu$.

Then $\map {\LL^p} \mu$ is a vector subspace of $\map \MM \Sigma$, the space of $\Sigma$-measurable functions.

In particular, it is a vector space.