Doubling measure

Remarks

Example: Lebesgue measure $\lambda$ on $\R^n$ is doubling with $C = 2^n$. Indeed $\lambda(B(x, r)) = \omega_n r^n$ with $\omega_n$ the volume of the unit ball, so the ratio is exactly $2^n$.

Non-example: the standard Gaussian measure $\gamma$ on $\R$ satisfies clause 1 but not clause 2. Let $\varphi(t) = e^{-t^2/2}/\sqrt{2\pi}$ be its density, which decreases on $[0, \infty)$, and let $x > 2$. Then $\gamma(B(x, 1)) \le 2 \varphi(x - 1)$ and $\gamma(B(x, 2)) \ge \frac12 \varphi(x - \frac32)$. So $\gamma(B(x,2)) / \gamma(B(x,1)) \ge \frac14 e^{(x - 5/4)/2}$, which is unbounded as $x \to \infty$.

Some authors drop clause 1 and only ask for clause 2 for balls of positive finite measure. A metric space with a doubling measure is a doubling metric space: each ball is covered by a bounded number of balls of half the radius.