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: $$\lim_{r\to 0}\int_{D(a,r)}f(t)dt=0?$$

  • 4
    $\begingroup$ Yes, by dominated convergence. $\endgroup$ – Angina Seng Oct 26 '18 at 16:45
  • $\begingroup$ In physics there are cases where one uses point charges to describe fields. A point charge is represented by the Dirac delta function. If you integrate over a sphere around the point charge, and then shrink the radius to $+0$, what you get is a constant term, not necessarily zero. $\endgroup$ – M. Wind Oct 26 '18 at 18:02
  • 1
    $\begingroup$ The question says explicitly "$f$ is a locally integrable function" ruling out the possibility that $f$ is a Dirac's delta. $\endgroup$ – Julián Aguirre Oct 26 '18 at 18:21
  • $\begingroup$ Lord Shark the Unknown: How would you deduce this from dominated convergence? I don't see it. $\endgroup$ – M. Rahmat Oct 27 '18 at 4:54
  • $\begingroup$ Why would it not be 0? Out of all the numbers out there, such as: 1, 15, 0, 92, 5673, -14i, why would it be anything other than 0 ? $\endgroup$ – Nike Dattani Oct 28 '18 at 22:08

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.