Integral of Integrable Function is Homogeneous

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

Let $f: X \to \overline \R$ be a $\mu$-integrable function.

Let $\lambda \in \R$.

Then:


 * $\displaystyle \int \lambda f \rd \mu = \lambda \int f \rd \mu$

where $\lambda f$ is the pointwise $\lambda$-multiple of $f$.