A question I recently saw while studying measures on $\mathbb R$:
Let $A$ be a set such that $A\cap K\neq \emptyset$ for any compact subset $K$ of real line of positive measure. Show that $A$ has infinite measure.
Intuitively, it feels as though this restriction on $A$ means that it must 'spread out' evenly throughout $\mathbb R$, but it is hard for me to formalize this in a way that helps tackle the problem. I know that $A$ must be dense in $\mathbb R$, though that doesn't really tell me much. I know that if $m(A)<\infty$, then $m(A-[-n,n])\rightarrow 0$ as $n\rightarrow \infty$ and it feels like this cannot be true under the conditions stated, but I can't figure out exactly why.
If $m(A) <\infty$ then $m(A^{c}) >0$. By regularity of Lebesgue measure there is a compact set $K \subset A^{c}$ with $m(K)>0$ and this contradicts the hypotheis. Ref. for regularity: Inner regularity of Lebesgue measure for $\mathbb R$