Definition:Absolutely Integrable Function

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Let $f$ be a real function.

$f \left({x}\right)$ 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 $\left[{\alpha \,.\,.\, \beta}\right] \subseteq S$ is bounded.