Given a measurable subset , we denote its
-dimensional Lebesgue measure by
.
Let be a bounded measurable set. Let
be a measurable function. For
, the level set
is defined as
.
The sets ,
,
and so on are defined by analogy. Then the distribution function of
is given by
.
This function is a monotonically decreasing function of and for
we have
while for
, we have
. Thus the range of
is the interval
.
Definition (Decreasing rearrangement). Let
be bounded and let
be a measurable function. Then the (unidimensional) decreasing rearrangement of
, denoted by
, is defined on
by
Essentially, is just the inverse function of the distribution function
of
. The following properties of the decreasing rearrangement are immediate from its definition.
Proposition 1. Let
where
is bounded. Then
is a nonincreasing and left-continuous function.
Proposition 2. The mapping
is non-decreasing, i.e. if
in the sense that
for all
, where
and
are real-valued functions on
then
.
We now see that is indeed a rearrangement of
.
Proposition 3. The function
and
are equimeasurable (i.e. they have the same distribution function), i.e. for all
.