Space of Integrable Functions is Vector Space

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

Let $\map {\LL^1} \mu$ be the space of real-valued $\mu$-integrable functions.

Then $\map {\LL^1} \mu$, endowed with pointwise $\R$-scalar multiplication and pointwise addition, forms a vector space over $\R$.