I'm looking for an example of a set $A\subset[0\,\pmb,\,1]$ such that the strict inequality$$0<\mathrm m(A\cap[a\,\pmb,\,b])<b-a$$holds for all real $a$ and $b$ with $0\leqslant a<b\leqslant1$, where $\mathrm m(X)$ denotes the Lebesgue measure of a measurable set $X\subset\Bbb R$.
My try: Start with the half-open interval $[0\,\pmb,\,1)$ and delete the middle half, so that what is left is $$[0\,\pmb,\,\tfrac14)\cup[\tfrac34\,\pmb,\,1).$$Next, delete the middle half of each quarter-length interval so formed, and centrally half-fill the gap of length half with an interval of length one quarter, to get $$[0\,\pmb,\,\tfrac1{16})\cup[\tfrac{3}{16}\,\pmb,\,\tfrac14)\cup[\tfrac38\,\pmb,\,\tfrac58)\cup[\tfrac34\,\pmb,\,\tfrac{13}{16})\cup[\tfrac{15}{16}\,\pmb,\,1).$$Continue in this way, so that, at each stage, we delete the middle half of each interval and fill in the middle half of each gap that was formed at the previous stage. However, because this process involves both deletions and additions (unlike e.g. the formation of the Cantor set, which involves only deletions), I'm not sure whether it converges (in some appropriate sense) to a set $A$ or, if $A$ is defined, whether it is measurable. The intention is to get $\mathrm m(A\cap[a\,\pmb,\,b])=\frac12(b-a).$
One issue with this approach is that there are many "Thomson lamps". In particular the midpoint $\frac12$ alternately appears or disappears at each stage, and the number of such "Thomsonian" points (e.g. $\frac18$ and $\frac78$ at the second stage) approximately trebles at each stage. These Thomsonian points alone are not a serious problem, since their inclusion or exclusion can be decided by some arbitrary rule—for example, by including the points with (lowest-terms) numerator of type $1$ modulo $4$ and excluding those of type $3$ modulo $4$. In any case, they form a set of measure zero. More seriously, though, there may be many more Thomsonian points, not arising as midpoints, comprising a set of sizeable measure—or even an unmeasurable set.
So my question is whether the task is possible in the first place and, if so, whether the above attempt can be engineered into working and also what might be a valid or simpler example.
