Integral of Positive Simple Function is Positive Homogeneous

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

Let $f: X \to \R, f \in \mathcal{E}^+$ be a positive simple function.

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

Then $I_\mu \left({\lambda \cdot f}\right) = \lambda \cdot I_\mu \left({f}\right)$, where:


 * $\lambda \cdot f$ is the pointwise $\lambda$-multiple of $f$
 * $I_\mu$ denotes $\mu$-integration

This can be summarized by saying that $I_\mu$ is positive homogeneous.

Proof
Remark that $\lambda \cdot f$ is a positive simple function by Scalar Multiple of Simple Function is Simple Function.

Let:


 * $f = \displaystyle \sum_{i \mathop = 0}^n a_i \chi_{E_i}$

be a standard representation for $f$.

Then we also have, for all $x \in X$:

and it is immediate from the definition that this yields a standard representation for $\lambda \cdot f$.

Therefore, we have:

as desired.

Also see

 * Integral of Positive Measurable Function is Positive Homogeneous, an extension of this result to positive measurable functions