# Simple function

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{{nofootnotes|date=January 2023}}
{{Short description|Function that attains finitely many values}}
In the [mathematical](/source/mathematics) field of [real analysis](/source/real_analysis), a '''simple function''' is a  [real](/source/real_number) (or [complex](/source/complex_number))-valued function over a subset of the [real line](/source/real_line), similar to a [step function](/source/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](/source/measurable_function), as used in practice.

A basic example of a simple function is the [floor function](/source/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](/source/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.) All [step function](/source/step_function)s are simple.

Simple functions are used as a first stage in the development of theories of [integration](/source/integral), such as the [Lebesgue integral](/source/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.

==Definition==

Formally, a simple function is a finite [linear combination](/source/linear_combination) of [indicator function](/source/indicator_function)s of [measurable set](/source/measurable_set)s. More precisely, let (''X'', Σ) be a [measurable space](/source/sigma-algebra). Let ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> ∈ Σ be a [sequence](/source/sequence) of disjoint measurable sets, and let ''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub> be a sequence of [real](/source/real_number) or [complex number](/source/complex_number)s. A ''simple function'' is a function <math>f\colon X \to \mathbb{C}</math> of the form

:<math>f(x)=\sum_{k=1}^n a_k {\mathbf 1}_{A_k}(x),</math>

where <math>{\mathbf 1}_A</math> is the [indicator function](/source/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](/source/algebra_over_a_field) over <math>\mathbb{C}</math>.

==Integration of simple functions==

If a [measure](/source/measure_(mathematics)) <math>\mu</math> is defined on the space <math>(X, \Sigma)</math>,  the [integral](/source/Lebesgue_integral) of a simple function <math>f\colon X \to \mathbb R</math> with respect to <math>\mu</math> is defined to be

:<math>\int_X f d \mu = \sum_{k=1}^na_k\mu(A_k),</math>
if all summands are finite.

==Relation to Lebesgue integration==
The above integral of simple functions can be extended to a more general class of functions, which is how the [Lebesgue integral](/source/Lebesgue_integral) is defined. This extension is based on the following fact.

: '''Theorem'''. Any non-negative [measurable function](/source/measurable_function) <math>f\colon X \to\mathbb{R}^{+}</math> is the [pointwise](/source/pointwise) limit of a monotonic increasing sequence of non-negative simple functions.

It is implied in the statement that the sigma-algebra in the co-domain <math>\mathbb{R}^{+}</math> is the restriction of the [Borel σ-algebra](/source/Borel_%CF%83-algebra) <math>\mathfrak{B}(\mathbb{R})</math> to <math>\mathbb{R}^{+}</math>. The proof proceeds as follows. Let <math>f</math> be a non-negative measurable function defined over the measure space <math>(X, \Sigma,\mu)</math>. For each <math>n\in\mathbb N</math>, subdivide the co-domain of <math>f</math> into <math>2^{2n}+1</math> intervals, <math>2^{2n}</math> of which have length <math>2^{-n}</math>. That is, for each <math>n</math>, define
:<math>I_{n,k}=\left[\frac{k-1}{2^n},\frac{k}{2^n}\right)</math> for <math>k=1,2,\ldots,2^{2n}</math>, and <math>I_{n,2^{2n}+1}=[2^n,\infty)</math>,

which are disjoint and cover the non-negative real line (<math>\mathbb{R}^{+} \subseteq \cup_{k}I_{n,k}, \forall n \in \mathbb{N}</math>).

Now define the sets 
:<math>A_{n,k}=f^{-1}(I_{n,k}) \,</math> for <math>k=1,2,\ldots,2^{2n}+1,</math>
which are measurable (<math>A_{n,k}\in \Sigma</math>) because <math>f</math> is assumed to be measurable.

Then the increasing sequence of simple functions 
:<math>f_n=\sum_{k=1}^{2^{2n}+1}\frac{k-1}{2^n}{\mathbf 1}_{A_{n,k}}</math> 

converges pointwise to <math>f</math> as <math>n\to\infty</math>. Note that, when <math>f</math> is bounded, the convergence is uniform.

==See also==
[Bochner measurable function](/source/Bochner_measurable_function)

== References ==
*{{aut|J. F. C. Kingman, S. J. Taylor}}. ''Introduction to Measure and Probability'', 1966, Cambridge.
*{{aut|[S. Lang](/source/Serge_Lang)}}. ''Real and Functional Analysis'', 1993, Springer-Verlag.
*{{aut|[W. Rudin](/source/Walter_Rudin)}}. ''Real and Complex Analysis'', 1987, McGraw-Hill.
*{{aut|H. L. Royden}}. ''Real Analysis'', 1968, Collier Macmillan.

{{DEFAULTSORT:Simple Function}}
Category:Real analysis
Category:Measure theory
Category:Types of functions

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Adapted from the Wikipedia article [Simple function](https://en.wikipedia.org/wiki/Simple_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Simple_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.