Set Function

set function

[′set ‚fəŋk·shən]
(mathematics)
A relation that assigns a value to each member of a collection of sets.

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 ʃM 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.

REFERENCES

Kolmogorov, A. N., and S. V. Fomin. Elementy teorii funktsii i funktsional’nogo analiza, 4th ed. Moscow, 1976.
Haimos, P. Teoriia mery. Moscow, 1953. (Translated from English.)
References in classic literature ?
They had spoken of it casually as something that would, as a matter of course, take place in the indefinite future, as, for instance, his promotion in the navy, in which he was now a padwar; or the set functions of the court of her grandfather, Tardos Mors, Jeddak of Helium; or Death.
Using the Line Set function, the beam can be moved up to 100 percent grade in a few seconds and then adjusted left or right to perform first day set in a short trench.
In order to maintain the level set function in the process of evolution stability, generally need to periodically as the symbolic distance function to initialize the level set function.
1) There is no primitive recursive set function that maps each RCF K to an integer part of K.
The main difference between these two kinds of approaches is that the geometric active contour introduces the level set function representing the evolving curve into the energy function.
Intelligent Parking Assist (with Easy Set function, Parking Space Confirmation function, and Handle Select function) is available.
The level set method which is devised by Osher and Sethian is a simple and versatile method for analyzing and computing the motion of interface, whose basic idea is that the interface is defined as the zero level set of level set function [phi] (X.
In the level set method, the contour is represented as zero level set of higher dimensional function called a level set function (LSF).
This is called a level set function, and it formulates the motion of the contour based on the evolution of the level set function.
In fact, the results described above can be seen as lifting known results for submodular set function maximization (in particular results by Nemhauser, Wolsey and Fisher [1978] and Minoux [1978]; see Krause and Golovin [2014] for a survey) to the sequential (partially observable) setting.
Customers also can select a Sanitize Devise Feature set function, which makes it possible to quickly invalidate data recorded on the magnetic disks.
They progress to maximum adjacency ordering and forest decompositions, minimum cuts, cut enumeration, cactus representations, extreme vertex sets, edge splitting, connectivity augmentation, source location problems, and semi- modular and posi-modular set function.