Consider a Lebesgue measurable set $E\subset\mathbb{R}$. Prove that the set $\{x\in\mathbb{R}:m(E\cap(x-k,x+k))\geq k, \forall k>0\}$ is Lebesgue measurable.
I am just a bit confused on where to begin. It looks like I can just apply the open set definition of measurability. That is, there exists an open set $O$ with $E\subset O$ and $m(O-E)\leq\epsilon$. But this would show that $E$ is measurable - and we already know that $E$ is. But wouldn't the set in question just be an open interval in $\mathbb{R}$, which we know is measurable? I feel like I am missing something quite simple....
Fix $k$. Consider the function $$f_k(x)=m(E\cap(x-k,x+k)).$$ This function is continuous: using that $m(A)-m(B)=m(A\setminus B)-m(B\setminus A)$ for measurable $A,B$ and assuming $x<y$, $$ |f_k(y)-f_k(x)|=|m(E\cap(x-k, y-k))-m(E\cap[(x+k, y+k))|\leq2|y-x|. $$ So $f_k$ is measurable, and $$ \{x\in\mathbb{R}:m(E\cap(x-k,x+k))\geq k, \forall k>0\}=\bigcap_kf_k^{-1}[k,\infty) $$ is measurable.