A singular measure with respect to a second measure is a measure concentrated
on a measurable set disjoint from one on which
the second measure is concentrated.
These measures are mutually
singular. For two complex measures 
and 
on the same measurable space 
, this means that 
for two disjoint sets 
and 
such that the following hold
for every measurable set 
.
1. The sets 
and 
are measurable.
2. The total variation of 
is concentrated on 
and that of 
on 
, i.e.,
 |
The relation of two complex measures being singular, written as 
, is 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 the measure 
concentrated
at 0, defined by 
when 
and 
otherwise. In general, a measure 
is concentrated on a
measurable set 
if 
for
every measurable set 
. For instance, the measure above
is concentrated at 0.
