Lebesgue Space is Vector Space

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

Let $\map {\LL^p} {X, \Sigma, \mu}$ be Lebesgue $p$-space on $\struct {X, \Sigma, \mu}$.

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

In particular, it is a vector space.