Example of positive measure compact set that contains no interval.

Question

Is Smith-Volterra-Cantor set[https://en.m.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set ] only example of positive Lebesgue measure compact set which does not contain interval?

Answer 1

$(1).$ Example. Let $S=\{s_n: n\in \Bbb N\}$ be any countable dense subset of $(0,1).$ Let $(t_n)_{n\in \Bbb N}$ be any sequence in $\Bbb R^+$ such that $\sum_{n\in \Bbb N}t_n<1/2.$

Let $U= \{(-t_n+s_n,\;t_n+s_n)\cap (0,1):n\in \Bbb N\}.$ Let $V=[0,1]\backslash \bigcup U.$ Since the measure of $\bigcup U$ cannot exceed $2\sum_{n\in \Bbb N}t_n ,$ which is less than $1,$ the measure of $V$ is not $0.$

And $V$ has empty interior because $[0,1]\backslash V\supset S$ and $S$ is dense in $[0,1].$

$(2).$ In general, if $V$ is compact with empty interior and positive measure let $a=\min V$ and $b=\max V .$ Then $(a,b)\backslash V=\bigcup F$ where $F$ is a countable family of pair-wise disjoint open sub-intervals of $(a,b).$ With $m$ denoting Lebesgue measure, we have $\sum_{f\in F}m(f)=m(\bigcup F) =(b-a)-m(V)<b-a.$

Note: In $(1)$, the family $U$ is not necessarily pair-wise disjoint, but $\bigcup U=\bigcup F$ where $F$ is a countable family of pair-wise disjoint open sub-intervals of $(0,1).$