Integral of Positive Simple Function is Positive Homogeneous

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

Let $f: X \to \R, f \in \EE^+$ be a positive simple function.

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

Then:
 * $\map {I_\mu} {\lambda \cdot f} = \map {\lambda \cdot I_\mu} f$

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:


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

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:

Hence the result.

Also see

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