Integral of Positive Measurable Function is Positive Homogeneous/Corollary

Corollary to Integral of Positive Measurable Function is Positive Homogeneous
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f : X \to \overline \R$ be a positive $\Sigma$-measurable function.

Let $\lambda \in \overline \R$ be an extended real number with $\lambda \ge 0$. Let $A \in \Sigma$.

Then:


 * $\ds \int_A \lambda f \rd \mu = \lambda \int_A f \rd \mu$

where:


 * $\lambda f$ is the pointwise $\lambda$-multiple of $f$
 * the integral sign denotes $\mu$-integration over $A$.

This can be summarized by saying that $\ds \int_A \cdot \rd \mu$ is positive homogeneous.

Proof
We have: