A measure universal set 
is a set such that every
measurable set 
of positive Lebesgue
measure contains a nontrivial affine copy of 
,
 |
where 
and 
.
These copies use affine transformations
with either sign of the scale, so they permit reflection
as well as translation and dilation.
Every nonempty finite set 
is measure universal. To see this, choose
a measurable subset 
of finite positive measure. Continuity of translations
in 
gives
 |
as 
,
where 
denotes Lebesgue measure. A point
of this intersection supplies an affine copy of
for sufficiently small nonzero 
. The Erdős
similarity conjecture asks whether every infinite set fails this property.
