I am currently stuck on a past paper question. It asks whether there exists a Lebesgue measurable set $D⊃[0,1]$ such that $D\neq [0,1]$ and $λ^{*}(D) =1?$ The only Lebesgue measurable set I have encountered with $λ^{*}(D) =1?$ in my module so far is $[0,1]∩\mathbb{Q}^{c}$ but $[0,1]\not\subset [0,1]∩\mathbb{Q}^{c}$. I have also tried to assume there does and doesn't but I can't seem to get anywhere. Any help would be appreciated.
2026-04-03 11:35:34.1775216134
Lebesgue measurable set with certain properties
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Just add any null set to $[0,1]$ which has elements outside of $[0,1]$, the measure will not change from that. $[0,1]\cup\{2\}$ will work, just as well as $[0,1]\cup\mathbb{Q}$.