Integral of Positive Simple Function is Positive Homogeneous

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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.

$\blacksquare$


Also see


Sources

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