I would like to know whether or not there exists a finite probability measure $\mu$ on $\mathbb R^2$ which has no atoms, but such that there exists an uncountable set $A\subset \mathbb S^1$, such that for every $v\in A$ there exists $x_v\in \mathbb R^2$ such that $\mu(x_v+v\mathbb R)>0$.

(A weaker requirement on $\mu$ would be that there exists an uncountable set of lines (i.e. affine $1$-dimensional subspaces of $\mathbb R^2$) $L$ such that for each $\ell \in L$ there holds $\mu(\ell)>0$, so proving that no positive finite Radon measure $\mu$ satisfies this property, would imply a negative answer to my question.)