Space of Integrable Functions is Vector Space

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

Let $\mathcal{L}^1 \left({\mu}\right)$ be the space of real-valued $\mu$-integrable functions.

Then $\mathcal{L}^1 \left({\mu}\right)$, endowed with pointwise $\R$-scalar multiplication and pointwise addition, forms a vector space over $\R$.