Function, Set

a function assigning a number to each set of some class.

For example, the length of an interval is a set function defined on the class of all intervals of a straight line; we speak in this case of an interval function. The integral

of an integrable function Φ(x) is also an interval function; here, the interval is the interval [α, β] of integration. Functions of regions in the plane or in space are also considered. Thus, for a given density distribution the mass contained in a region is a function of the region.

The concept of a region function is a more flexible tool for describing physical phenomena than the concept of a point function because the region function concept permits the consideration of cases where the density of a physical quantity is infinite at isolated points, such as point sources. Moreover, the concept corresponds better to the reality of physical experiments, where one observes not a point function but its average over a small region.

The development of the concept of a set function was associated with the construction of the Lebesgue theory of integration. In that theory one has to consider not only functions of a region but also functions of an arbitrary measurable set. One of the first examples of such a set function was the Lebesgue measure μ(E) of a measurable set E (seeMEASURE OF A SET). This set function is completely additive—that is, the measure of the sum of any finite or countable collection of disjoint measurable sets is equal to the sum of the measures of the sets. Lebesgue measure is one of the many measures that are nonnegative completely additive set functions defined on appropriate classes of sets. Such set functions occur in the general theory of integration.

A set function f(E) is said to be absolutely continuous with respect to a measure μ, if f(E) = 0 when μ(E) = 0. Thus, the Lebesgue integral ʃ_M ɸ(x) dx of a given summable function φ(x) over a set M is a completely additive function of M that is absolutely continuous with respect to Lebesgue measure. Conversely, any completely additive absolutely continuous set function can be represented as the Lebesgue integral of a summable function φ(x).

An important example of a set function is a probability distribution.