I have the following exercise from my measure theory class, and to be honest I’m quite confused. It goes as follows:
For every $x\in\mathbb{R}^3$ we call the needle with basis $x$ the horizontal line segment $$ \{x + (a,0,0) : a\in(0,1]\}. $$ In particular, $x$ is not a member of the needle at $x$. If $K\subseteq\mathbb{R}^3$ has the property that, for each $x \in K$, the whole needle with basis $x$ lives outside of $K$ (meaning it’s in ${K}^\complement$), show that $\lambda(K)=0$. Here $\lambda$ is the Lebesgue measure on $\mathbb R^3$.
I couldn’t find anything about this needle and I would like to get some intuition about the problem, so I could use some hints and not the full solution at least at first.
Thanks in advance.