Integral of Positive Simple Function is Increasing
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Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f, g: X \to \R$, $f, g \in \EE^+$ be positive simple functions.
Suppose that:
- $f \le g$
where $\le$ denotes pointwise inequality.
Then:
- $\map {I_\mu} f \le \map {I_\mu} g$
where $I_\mu$ denotes $\mu$-integration
This can be summarized by saying that $I_\mu$ is an increasing mapping.
Proof
Note that:
- $g - f \ge 0$
From Scalar Multiple of Simple Function is Simple Function and Pointwise Sum of Simple Functions is Simple Function, we then have that:
- $g - f \in \EE^+$
Write:
- $g = f + \paren {g - f}$
From Integral of Positive Simple Function is Additive, we then have:
- $\map {I_\mu} g = \map {I_\mu} f + \map {I_\mu} {g - f}$
Since:
- $\map {I_\mu} {g - f} \ge 0$
we obtain:
- $\map {I_\mu} f \le \map {I_\mu} g$
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $9.3 \ \text{(iv)}$