Scalar Multiple of Integrable Function is Integrable Function

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

Let $f: X \to \overline \R$ be a $\mu$-integrable function, and let $\lambda \in \R$.

Then $\lambda f: X \to \overline \R$, the pointwise $\lambda$-multiple of $f$, is also $\mu$-integrable.

That is, the space of integrable functions $\mathcal{L}^1_{\overline \R}$ is closed under pointwise $\R$-scalar multiplication.