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
.