I was reading Construction of a Borel set with positive but not full measure in each interval and would like to extend the question:

Let $\ell:\mathscr{B}(\mathbb{R})\to[0,\infty]$ be the Lebesgue measure defined over the Borel sigma algebra on $\mathbb{R}$. Consider for some $S\in\mathscr{B}(\mathbb{R})$ the function $f_s:\mathbb{R}\to[0,1]$ defined so: $$f_s(x) = \ell(S\cap[x,x+1])$$

Does there exist some $S\in\mathscr{B}(\mathbb{R})$ s.t. $\ell(S)=\infty$ and $\ell(S^c)=\infty$ and $f_s$ is constant? My feeling is a strong no, but I'm having great difficulty coming any close to a proof.

P.S. This is my first time posting; pls let me know how I can improve.


Just pick $S = \bigcup_{n = -\infty}^\infty \big[n, n+\frac{1}{2}\big)$. Then $S$ and $S^c$ both have infinite measure and $S \cap [x, x+1]$ consists of one or two intervals with combined length $\frac{1}{2}$ for all $x \in \mathbb{R}$. Therefore $f_s \equiv \frac{1}{2}.$


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