Integral of Positive Measurable Function is Positive Homogeneous

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Theorem

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

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

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


Then:

$\displaystyle \int \lambda f \rd \mu = \lambda \int f \rd \mu$

where:

$\lambda f$ is the pointwise $\lambda$-multiple of $f$
The integral sign denotes $\mu$-integration


This can be summarized by saying that $\displaystyle \int \cdot \rd \mu$ is positive homogeneous.


Proof


Also see


Sources