A metric space $X$ with metric $d$ is said to be doubling on $\Bbb R^2$ if there is some constant $C > 0$ such that for any $x \in X$ and $r > 0$, the Euclidean ball $B(x, r) = \{y:|x − y| < r\}$ may be contained in a union of no more than $C$ many balls with radius $r/2$.
That is:
$$B(x, 2r) < C \cdot B(x,r).$$
Can we prove that any such measure gives measure zero to a straight line $L$?