Measure which does not grow faster than Lebesgue

Question

Is there an example of a measure $\mu$ on $\mathbb{R}$ which is not absolutely continuous with respect to Lebesgue measure such that $\mu[\mathbb{R}]=+\infty$ but $$\limsup_{a\to +\infty}\frac{a}{\mu[-a,a]}<+\infty.$$

Answer 1

Let $\mu$ be counting measure on $\mathbb{Z}$; that is,

$$\mu(E) = |E \cap \mathbb{Z}|$$

It's easy to verify this is a measure which is not absolutely continuous with respect to Lebesgue measure; furthermore,

$$\mu([-a,a]) = 1 + 2 \lfloor a \rfloor$$

(by symmetry, together with $0$ in the middle), so that

$$\limsup_{a \to \infty} \frac{a}{\mu([-a,a])} = \frac 1 2$$

Notice how this example can be easily adapted (by either considering where the point masses are, or the weights at each point) so that the $\limsup$ is equal to any given positive number.

Alternatively, if $\mu$ is the measure which is $0$ for $\emptyset$ and $\infty$ otherwise, then the limit supremum is zero.