Integral of Positive Measurable Function is Monotone
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Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.
Let $f,g: X \to \overline{\R}$, $f,g \in \mathcal{M}_{\overline{\R}}^+$ be positive measurable functions.
Suppose that $f \le g$, where $\le$ denotes pointwise inequality.
Then:
- $\displaystyle \int f \, \mathrm d\mu \le \int g \, \mathrm d\mu$
where the integral sign denotes $\mu$-integration.
This can be summarized by saying that $\displaystyle \int \cdot \, \mathrm d\mu$ is monotone.
Proof
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $9.8 \ \text{(iv)}$, $\S 9$: Problem $2$