Integral of Positive Measurable Function is Monotone

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.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $9.8 \ \text{(iv)}$, $\S 9$: Problem $2$