#jsDisabledContent { display:none; } My Account |  Register |  Help

# Simple function

Article Id: WHEBN0001890594
Reproduction Date:

 Title: Simple function Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Simple function

In the mathematical field of real analysis, a simple function is a real-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently 'nice' that using them makes mathematical reasoning, theory, and proof easier. For example simple functions attain only a finite number of values. Some authors also require simple functions to be measurable; as used in practice, they invariably are.

A basic example of a simple function is the floor function over the half-open interval [1,9), whose only values are {1,2,3,4,5,6,7,8}. A more advanced example is the Dirichlet function over the real line, which takes the value 1 if x is rational and 0 otherwise. (Thus the "simple" of "simple function" has a technical meaning somewhat at odds with common language.) Note also that all step functions are simple.

Simple functions are used as a first stage in the development of theories of integration, such as the Lebesgue integral, because it is easy to define integration for a simple function, and also, it is straightforward to approximate more general functions by sequences of simple functions.

## Contents

• Definition 1
• Properties of simple functions 2
• Integration of simple functions 3
• Relation to Lebesgue integration 4
• References 5

## 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, ..., An ∈ Σ be a sequence of disjoint measurable sets, and let a1, ..., an be a sequence of real or complex numbers. A simple function is a function f: X \to \mathbb{C} of the form

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

where {\mathbf 1}_A is the indicator function of the set A.

## Properties of simple functions

The sum, difference, and product of two simple functions are again simple functions, and multiplication by constant keeps a simple function simple; hence it follows that the collection of all simple functions on a given measurable space forms a commutative algebra over \mathbb{C}.

## Integration of simple functions

If a measure μ is defined on the space (X,Σ), the integral of f with respect to μ is

\sum_{k=1}^na_k\mu(A_k),

if all summands are finite.

## Relation to Lebesgue integration

Any non-negative measurable function f\colon X \to\mathbb{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 the measure space (X, \Sigma,\mu) as before. For each n\in\mathbb N, subdivide the range of f into 2^{2n}+1 intervals, 2^{2n} of which have length 2^{-n}. For each n, 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).

(Note that, for fixed n, the sets I_{n,k} are disjoint and cover the non-negative real line.)

Now 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 n\to\infty. Note that, when f is bounded, the convergence is uniform. This approximation of f by simple functions (which are easily integrable) allows us to define an integral f itself; see the article on Lebesgue integration for more details.

## 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.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.