A set 
is said to be nowhere dense if the interior
of the set closure of 
is the empty set. For example,
the Cantor set is nowhere dense.
There exist nowhere dense sets of positive measure. For example, enumerating the rationals in ![[0,1]](/images/equations/NowhereDense/Inline3.svg)
as 
and choosing an open interval 
of length 
containing 
for each 
, then the union of these intervals has measure at most 1/2.
Hence, the set of points in ![[0,1]](/images/equations/NowhereDense/Inline9.svg)
but not in any of 
has measure at least 1/2, despite being nowhere dense.
