Integral of Positive Measurable Function is Positive Homogeneous

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

Let $f: X \to \R, f \in \mathcal{M}_{\overline{\R}}^+$ be a positive measurable function.

Let $\lambda \in \R_{\ge 0}$ be a positive real number.

Then:


 * $\displaystyle \int \lambda f \, \mathrm d\mu = \lambda \int f \, \mathrm d\mu$

where:


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

This can be summarized by saying that $\displaystyle \int \cdot \, \mathrm d\mu$ is positive homogeneous.

Also see

 * Integral of Positive Simple Function is Positive Homogeneous, a similar result for positive simple functions.