# 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 $x \in X$:

 $\ds \map {\lambda \cdot f} x$ $=$ $\ds \lambda \sum_{i \mathop = 0}^n a_i \map {\chi_{E_i} } x$ $\ds$ $=$ $\ds \sum_{i \mathop = 0}^n \paren {\lambda a_i} \map {\chi_{E_i} } x$ Summation is Linear

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

Therefore, we have:

 $\ds \map {\lambda \cdot I_\mu} f$ $=$ $\ds \lambda \sum_{i \mathop = 0}^n a_i \map \mu {E_i}$ Definition of $\mu$-Integration $\ds$ $=$ $\ds \sum_{i \mathop = 0}^n \paren {\lambda a_i} \map \mu {E_i}$ Summation is Linear $\ds$ $=$ $\ds \map {I_\mu} {\lambda \cdot f}$ Definition of $\mu$-Integration

Hence the result.

$\blacksquare$