NationMaster - Encyclopedia: Simple function

In mathematics, especially in mathematical analysis, a simple function is a measurable function whose range is finite. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ... In mathematics, the range of a function is the set of all output values produced by that function. ...

Simple functions are used as a first stage in the development of theories of integration, such as the Lebesgue integral, due to the fact that it is very easy to create a definition of an integral for a simple function, and also, it is straightforward to approximate more general functions by sequences of simple functions. In calculus, the integral of a function is an extension of the concept of a sum. ... In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ...

1 Examples
2 Definition
3 Properties of simple functions
4 Integration of simple functions
5 References

Examples

Consider a function of a real variable x. Define f(x) = 0 when x is 0, f(x)=−1 when x is negative, and f(x)=1 when x is positive. Then f is a simple function, since its range is {-1, 0, 1}, which is a finite set, and one can check that this function is measurable on the usual space of Lebesgue measurable sets. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ...

Another example is the indicator function of the rational numbers, which takes the value 1 on the measurable set $mathbb{Q}$ and the value 0 on the measurable set $mathbb{R} setminus mathbb{Q}.$ In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, a measure is a function that assigns a number, e. ...

Definition

Formally, a simple function is a finite linear combination of indicator functions of measurable sets. More precisely, let (X, Σ) be a measurable space. Let A₁, ..., A_n ∈ Σ be a sequence of measurable sets, and let a₁, ..., a_n be a sequence of real or complex numbers. A simple function is a function of the form In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

$f(x)=sum_{k=1}^n a_k {mathbf 1}_{A_k}(x).$

Properties of simple functions

By definition, sum, difference, and product of two simple functions is again a simple function, as well multiplication by constant, hence it follows that the collection of all simple functions forms a commutative algebra over the complex field.

For the development of a theory of integration, the following result is important. Any non-negative measurable function $fcolon X tomathbb{R}^{+}$ is the pointwise limit of a monotonic increasing sequence of non-negative simple functions. Indeed, let $f$ be a non-negative measurable function defined over a measure space $(Omega, {mathcal F},mu)$ . For each $ninmathbb N$ , we subdivide the range of $f$ into $2 2 n + 1$ intervals of length $2 - n$ . We set $I_{n,k}=left[frac{k-1}{2^n},frac{k}{2^n}right)$ for $k=1,2,ldots,2^{2n}$ and $I_{n,2^{2n}+1}=[2^n,infty]$ . We define the measurable sets $A n, k = f - 1 (I n, k)$ for $k=1,2,ldots,2^{2n}+1$ . Then the increasing sequence of simple functions $f_n=sum_{k=1}^{2^{2n}+1}frac{k-1}{2^n}{mathbf 1}_{A_{n,k}}$ converges pointwise to $f$ as $ntoinfty$ . In mathematics, a measure is a function that assigns a number, e. ...

Note that when $f$ is bounded the convergence is uniform.

Integration of simple functions

If a measure μ is defined on the space (X,Σ), the integral of f with respect to μ is In mathematics, a measure is a function that assigns a number, e. ... In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ...

$sum_{k=1}^na_kmu(A_k),$

if all summands are finite.

References

J. F. C. Kingman, S. J. Taylor. Introduction to Measure and Probability, 1966, Cambridge.
S. Lang. Real and Functional Analysis, 1993, Springer-Verlag.
W. Rudin. Real and Complex Analysis, 1987, McGraw-Hill.
H. L. Royden. Real Analysis, 1968, Collier Macmillan.

Categories: Mathematical analysis stubs | Measure theory

Results from FactBites:

simple function: Definition and Much More from Answers.com (434 words)
	In mathematics, especially in mathematical analysis, a simple function is a measurable function whose range is finite.
	Simple functions are used as a first stage in the development of theories of integration, such as the Lebesgue integral, due to the fact that it is very easy to create a definition of an integral for a simple function, and also, it is straightforward to approximate more general functions by sequences of simple functions.
	Formally, a simple function is a finite linear combination of indicator functions of measurable sets.

Procedural Dylan -- A Simple Function (1498 words)
	Upon reflection, one might come to the conclusion that the name is unnecessary as documentation, since the function name should describe the meaning of the return value.
	The last important detail to notice is that, aside from the issue of spaces, the expression used to calculate the distance is the same in both Dylan and Pascal.
	Another way to rewrite the distance function to only do the substractions once would be to abstract out the squaring operation as a local function.

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Contents

Definition

Properties of simple functions

Integration of simple functions

References