Integral of Integrable Function is Homogeneous

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Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $f: X \to \overline{\R}$ be a $\mu$-integrable function, and let $\lambda \in \R$.


Then:

$\displaystyle \int \lambda f \, \mathrm d \mu = \lambda \int f \, \mathrm d \mu$

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


Proof


Sources