When does the limit of the mean values of a function around a point approach the value of the function at that point ? We can prove it if the function is continuous. But are there general classes of functions for which this holds ?

In precise language , what is the most general class of functions for which the following hold ?

$\frac{1}{n\alpha(n)\epsilon^{n-1}}\int_{\partial B(x,\epsilon)}f(y)dS(y) \rightarrow f(x)$ as $\epsilon \rightarrow 0 $

  • $\begingroup$ If the function is harmonic, this comes straight from the definition, perhaps more context is needed $\endgroup$ – Ellya Jun 11 '14 at 8:14
  • $\begingroup$ yes but f(y) is not harmonic it comes from poisson equation delta u = -f.how it comes from definition? $\endgroup$ – Raja Sekar Jun 11 '14 at 8:27
  • $\begingroup$ if it were harmonic, the mean integral over a ball centred at $x$ is the function evaluated at $x$. $\endgroup$ – Ellya Jun 11 '14 at 8:31
  • $\begingroup$ ok sorry mean value property.again sorry sometimes it happens.but can you please check this out and give me some explanation.(math.stackexchange.com/questions/828836/…) $\endgroup$ – Raja Sekar Jun 11 '14 at 8:37
  • $\begingroup$ Pointwise it can be absolutely horrible, note that the limit holds if you replace $f(y)$ by $f(y) + g(y)$ where around every ball $\partial B(x,\epsilon)$ the integral of $g$ vanishes. In particular, $g(y)$ doesn't even need to be Lebesgue integrable on $B(x,1)$. If you want this to hold everywhere, then the question may get more interesting. $\endgroup$ – Willie Wong Jun 23 '14 at 11:16

The answer to this is the content of the Lebesgue differentiation theorem.

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    $\begingroup$ It doesn't look to me like the original question is asking about functions for which this holds almost everywhere. I guess as long as the derivative of the integral exists everywhere for a Lebesgue integrable function $f$, then what you can say is that it is almost everywhere equal to a unique representative function which has the property OP asks for. $\endgroup$ – Dustan Levenstein Jun 23 '14 at 6:06
  • $\begingroup$ Thank you for pointing out.The theorem only proves the equality a.e for locally integrable functions.I'm trying to understand this theorem. Is there any detailed exposition of this theorem apart from the wiki page ? $\endgroup$ – Srinivas K Jun 23 '14 at 7:20
  • $\begingroup$ Srinivas it is just mean value theorem $\endgroup$ – Raja Sekar Jun 23 '14 at 12:08
  • $\begingroup$ Can you please elaborate ? $\endgroup$ – Srinivas K Jun 23 '14 at 13:34
  • $\begingroup$ This has been used even before introducing MVT in Page 25 of Evans when there are no restrictions on $f$. $\endgroup$ – Srinivas K Jun 28 '14 at 12:13

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