From the definition of a doubling measure on wikipedia:

"A measure on a metric space X is said to be doubling if the measure of any ball is approximately the measure of its double, or more precisely, if there is a constant $C > 0$ such that

$\mu(B(x,2r))\leq C\mu(B(x,r))$

for all x in X and r > 0. In this case, we say μ is C-doubling.


A simple example of a doubling measure is Lebesgue measure on a Euclidean space"

This doesn't make sense to me. The description says that "the measure of any ball is approximately the measure of its double", but that doesn't seem to be true for the $2d$ Lebesgue measure, ie area.

I'm just confused and I have a limited understanding of measure theory. Could someone please clarify?

  • $\begingroup$ In the Euclidean $n$-space: $(2r)^n \le 2^n r^n$, so you can take $C=2^n$ in the definition. $\endgroup$ – Moishe Kohan Nov 29 '13 at 19:52

"Approximately equal to" here means "bounded from above and below by some constant multiple of".

The Lebesgue measure $\lambda_2$ on $\mathbb R^2$ has the doubling property since, for every $x$ in $\mathbb R^2$ and every $r\geqslant0$, $\lambda_2(B(x,2r))=4\pi r^2=4\lambda_2(B(x,r))$.

More generally, for every $d\geqslant1$, the Lebesgue measure $\lambda_d$ on $\mathbb R^d$ has the doubling property with constant $C=2^d$.

  • $\begingroup$ Ok that's very helpful. Thank you. I might edit the wiki entry to make some note of this. $\endgroup$ – Bryan Glazer Nov 29 '13 at 20:03
  • $\begingroup$ Not sure that adding this to the WP page is absolutely necessary (but I am not an expert in the edition of WP pages). $\endgroup$ – Did Nov 29 '13 at 20:05

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