# The product of a bounded and an absolutely integrable function is absolutely integrable

There is a theorem in Fourier Series where in the proof they have used the following result:
The product of a bounded and an absolutely integrable function is absolutely integrable
But the proof of this result has not been told to us. In this previously answered question , I have seen the proof but was not able to understand properly because of the terms "almost everywhere". Actually, I'm merely a beginner in this topic of Integrability and Fourier Series. Can anyone provide me with a more elaborate form of this proof for a better understanding. I will be awfully greatful for this !
Thankyou.

• The statement is false for Riemann integrability. It is obvious for Lebesgue integrability if the bounded function is also measurable. – Kavi Rama Murthy Sep 16 at 5:02
• Absolutely integrable functions are only Lebesgue integrable I guess ? – Esha Sep 16 at 5:09