# Pointwise Difference of Measurable Functions is Measurable

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## Theorem

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

Let $f, g: X \to \overline \R$ be $\Sigma$-measurable functions.

Assume that the pointwise difference $f - g: X \to \overline \R$ is well-defined.

Then $f - g$ is a $\Sigma$-measurable function.

## Proof

We have the apparent identity:

- $f - g = f + \paren {-g}$

By Negative of Measurable Function is Measurable, $-g$ is a measurable function.

Hence so is $f - g$, by Pointwise Sum of Measurable Functions is Measurable.

$\blacksquare$

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $8.10$