Integral of Positive Simple Function is Increasing

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Theorem

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

Let $f, g: X \to \R$, $f, g \in \mathcal{E}^+$ be positive simple functions.


Suppose that:

$f \le g$

where $\le$ denotes pointwise inequality.


Then:

$I_\mu \left({f}\right) \le I_\mu \left({g}\right)$

where $I_\mu$ denotes $\mu$-integration


This can be summarized by saying that $I_\mu$ is an increasing mapping.


Proof


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