# Are there any measurable, uncountable sets dense in [0,1] with Lesbegue measure less than 1?

Sorry tried to formulate this question the other day, got myself into a bit of a pickle.

(By dense in [0,1] I mean there is an element in our set arbitrarily close to every point in [0,1]. If that isn't what dense in [0,1] means apologies.)

Some examples I have considered;

• The rationals (Countable, so no)
• The real line with the rationals removed (Measure 1 so no)
• Cantor set doesn't have anything within 1/6 of 0.5, right?

If not, does anyone know of a proof of the result?

Sure. Take $[0,\frac{1}{2}]\cup ([\frac{1}{2},1]\cap\mathbb{Q}])$. This set is dense, uncountable and has Lebesgue measure $\frac{1}{2}$.
• @user70147: You could do something like this. Enumerate the rationals in $[0,1]$ by $\{q_{k}:k\in\mathbb{N}\}$, and take $\frac{1}{2^{k+2}}$ radii ball around each $q_{k}$. Take the union of these balls intersected with $[0,1]$. It has measure at most $\frac{1}{2}$ and bigger than $0$, it is dense, and what can you say about its intersection with every interval on $[0,1]$? (since every interval contains rationals) Dec 8, 2013 at 16:31
• Sure, just replace the appearance of $\mathbb Q$ in the response with a set that’s uncountable and of measure zero, such as (I think) the set of reals with only finitely many $7$’s in the decimal expansion. Dec 8, 2013 at 16:33