Definition:Absolutely Integrable Function

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Let $f$ be a real function.

$\map f x$ is absolutely integrable on $S \subseteq \R$ if and only if its absolute value of $f$ is an integrable function on $S$.

That is, if and only if the definite integral of the absolute value of $f$ over any interval $\openint \alpha \beta \subseteq S$ is bounded.