The ordinary Cantor set is obtained by removing the middle third of a unit line segment and iterating. The resulting set of points is uncountable, nowhere dense and has zero measure.
It transpires that the construction can be modified to give a set with non-zero measure, the so-called fat Cantor set. Note that in the previous construction, we remove intervals whose total length sums to 1 (namely 1/3 + 2/9 + 4/27 + 8/81 + …). If, instead, we remove smaller intervals so that the sum converges to a value below 1, then the set left over will have non-zero measure. An example of this is the Smith-Volterra-Cantor set, with measure 1/2.
Indeed, by a similar construction, we can get nowhere dense sets with measure arbitrarily close to (but not equal to) 1. This result is so counter-intuitive that the fat Cantor set was proclaimed ‘bizarre mathematical object of the week’ by Ben Elliott et al.