Limit of an integral over a ball as the radius of the ball goes to zero 2

Question

Let $D(a,r)$ be an open ball in $\mathbb{R}^{k}$ ($ k\geq1 $), and $f$ locally integrable function in $\mathbb{R}^{k}$. Do we have: $$\lim_{r\to 0}\int_{D(a,r)}f(t)dt=0?$$

Answer 1

The answer is YES. This is a classic result called "absolute continuity of The Lebesgue integral", for example see rmh52 (https://math.stackexchange.com/users/35961/rmh52), Absolute continuity of the Lebesgue integral, URL (version: 2017-02-03): Absolute continuity of the Lebesgue integral