# Integral of Positive Measurable Function is Positive Homogeneous

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## Contents

## Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f: X \to \R, f \in \mathcal M_{\overline \R}^+$ be a positive measurable function.

Let $\lambda \in \R_{\ge 0}$ be a positive real number.

Then:

- $\displaystyle \int \lambda f \rd \mu = \lambda \int f \rd \mu$

where:

- $\lambda f$ is the pointwise $\lambda$-multiple of $f$
- The integral sign denotes $\mu$-integration

This can be summarized by saying that $\displaystyle \int \cdot \rd \mu$ is positive homogeneous.

## Proof

## Also see

- Integral of Positive Simple Function is Positive Homogeneous, a similar result for positive simple functions.

## Sources

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