Function Simple iff Positive and Negative Parts Simple

Theorem

Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $g: X \to \overline{\R}$ be an extended real-valued function.

Then $g$ is a simple function if and only if its positive part $g^+$ and negative part $g^-$ are simple functions.

Proof

Necessary Condition

Suppose $g$ is a simple function.

By Positive Part of Simple Function is Simple Function, so is $g^+$.

By Negative Part of Simple Function is Simple Function, so is $g^-$.

$\Box$

Sufficient Condition

Suppose $g^+$ and $g^-$ are simple functions.

From Difference of Positive and Negative Parts:

$g = g^+ - g^-$

Hence $g$ is simple, by Pointwise Difference of Simple Functions is Simple Function.

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Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 8$: Problem $5$