Integral of Positive Simple Function is Positive Homogeneous

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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 $x \in X$:

\(\displaystyle \lambda \cdot f \left({x}\right)\) \(=\) \(\displaystyle \lambda \sum_{i \mathop = 0}^n a_i \chi_{E_i} \left({x}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 0}^n \left({\lambda a_i}\right) \chi_{E_i} \left({x}\right)\) Summation is Linear

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


Therefore, we have:

\(\displaystyle \lambda \cdot I_\mu \left({f}\right)\) \(=\) \(\displaystyle \lambda \sum_{i \mathop = 0}^n a_i \mu \left({E_i}\right)\) Definition of $\mu$-integration
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 0}^n \left({\lambda a_i}\right) \mu \left({E_i}\right)\) Summation is Linear
\(\displaystyle \) \(=\) \(\displaystyle I_\mu \left({\lambda \cdot f}\right)\) Definition of $\mu$-integration

as desired.

$\blacksquare$


Also see


Sources