# 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

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $9.3 \ \text{(iv)}$