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 rearrangementof , 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.

Proposition1. Let where is bounded. Then is a nonincreasing and left-continuous function.

Proposition2. 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 .

Proposition3. The function and areequimeasurable(i.e. they have the same distribution function), i.e. for all.