I've been trying to find some notes on the following statement: Let $f:(a,b] \to \mathbb{R}$, $f\geq 0$, and $f\in\mathcal{R}[a+\epsilon , b]$ for any $\epsilon>0$. Then $\int_a^bf=\lim_{\epsilon \to 0} \int_{a+\epsilon}^b f<\infty$ if and only if $f\in L^1[a,b]$ and $\int_{[a,b]}=\int_a^bf$.
I have posted a proof a while back but did not get much discussion on it and so I thought I would create this post in hopes of finding a correct proof for such a statement. I apologize if this goes against some rule/guideline about such posts. Thank you.