Distributions may be defined on all binary strings
by first selecting lengths and then selecting strings
of that length. Although it is mathematically impossible to
select strings with equal chance
from an infinite sample space,
strings of the same length can be selected with equal likelihood.
It is also impossible to select integers
from 
with the same probability,
but one can select
an integer with a probability close to being ``uniform.''
A p-time computable
distribution 
on 
is called uniform if
for all 
, 
,
where 
and there is a polynomial 
such that
for all but finitely many 
, 
.
It is important to note that for the purpose of
proving completeness
results, is the only requirement needed
since domination allows a polynomial factor,
and so some longer
strings can certainly be given more weights than shorter ones.
Levin [Lev86] used 
for 
for notational
convenience (normalized by dividing by
), and
is often referred to as
the default uniform distribution.