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.
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Before, you had to either memorise or set function keys to capture screenshots on a Mac.
In addition, keeping numerical stability is necessary and essential in implementing the level set methods, which implies that the corresponding level set function (LSF) can't be very flat or steep near the zero level set.
In this method, the implicit level set function is interpolated by radial basis functions (RBFs), the H-J PDE is converted into a series of ordinary differential equations which are easier to handle, and the propagation of boundaries is driven by updating interpolated coefficients using mathematical programming methods.
The level set function [empty set] introduced by Osher and Sethian [18] is an efficient numerical method for solving curve evolution.
The basic idea of the level set is to consider the boundary surface as the zero level set of a function [psi] (called the level set function) in the higher dimension space, and the speed of the boundary surface also expands to the level set function in higher dimension.
We now prove the following properties for the set function E [right arrow] [C.sub.[phi](*)].
Thirdly, a level set function update strategy called SFM [21] is used to minimize the number of pixels of the level set function that need to be updated in each iteration.
-1 [less than or equal to] [phi](a) [less than or equal to] 1: level set function in which [phi](a)=0 specifies the interface
In order to indicate the belief degree that an event happens, Liu [1] defined a definition of uncertain measure as a set function satisfying normality, duality, subadditivity, and product axiom.
The level set function is the signed MD from the pixel to the curve.