Two complex measures 
and 
on a measure space 
, are mutually singular if they are supported
on different subsets. More precisely, 
where 
and 
are two disjoint sets such
that the following hold for any measurable set 
,
1. The sets 
and 
are measurable.
2. The total variation of 
is supported on 
and that of 
on 
, i.e.,
 |
The relation of two measures being singular, written as 
, is plainly symmetric. Nevertheless, it is sometimes
said that "
is singular with respect to 
."
A discrete singular measure (with respect to Lebesgue measure on the reals) is a measure 
supported at 0, say 
iff 
. In general, a measure 
is concentrated on a subset 
if 
. For instance, the measure
above is concentrated at 0.
