Integral of Integrable Function is Homogeneous

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Theorem

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

Let $f: X \to \overline \R$ be a $\mu$-integrable function.

Let $\lambda \in \R$.


Then:

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

where $\lambda f$ is the pointwise $\lambda$-multiple of $f$.


Proof


Sources