Examples of uncountable sets with zero Lebesgue measure I would like examples of uncountable subsets of $\mathbb{R}$ that have zero Lebesgue measure and are not obtained by the Cantor set.
Thanks.

Let $(r_n)_{n\in\mathbb{N}}$ be a dense sequence in $\mathbb{R}$ (it could, for example, be an enumeration of the rationals). For $k \in \mathbb{N}$, let
$$U_k = \bigcup_{n\in\mathbb{N}} (r_n - 2^{-(n+k)},\, r_n + 2^{-(n+k)}).$$
$U_k$ is a dense open set with Lebesgue measure $\leqslant 2^{2-k}$, thus
$$N = \bigcap_{k\in\mathbb{N}} U_k$$
is a set of the second category (Baire), hence uncountable, and has Lebesgue measure $0$.
