# Integral of Positive Simple Function is Additive

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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.

Then $I_\mu \left({f + g}\right) = I_\mu \left({f}\right) + I_\mu \left({g}\right)$, where:

- $f + g$ is the pointwise sum of $f$ and $g$
- $I_\mu$ denotes $\mu$-integration

This can be summarized by saying that $I_\mu$ is additive.

## Proof

## Sources

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