# Simple function

__: Function that attains finitely many values__

**Short description**In the mathematical field of real analysis, a **simple function** is a real (or complex)-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.) 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.

## 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 *A*_{1}, ..., *A*_{n} ∈ Σ be a sequence of disjoint measurable sets, and let *a*_{1}, ..., *a*_{n} be a sequence of real or complex numbers. A *simple function* is a function [math]\displaystyle{ f: X \to \mathbb{C} }[/math] of the form

- [math]\displaystyle{ f(x)=\sum_{k=1}^n a_k {\mathbf 1}_{A_k}(x), }[/math]

where [math]\displaystyle{ {\mathbf 1}_A }[/math] 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 [math]\displaystyle{ \mathbb{C} }[/math].

## Integration of simple functions

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

- [math]\displaystyle{ \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 is defined. This extension is based on the following fact.

**Theorem**. Any non-negative measurable function [math]\displaystyle{ f\colon X \to\mathbb{R}^{+} }[/math] is the 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]\displaystyle{ \mathbb{R}^{+} }[/math] is the restriction of the Borel σ-algebra [math]\displaystyle{ \mathfrak{B}(\mathbb{R}) }[/math] to [math]\displaystyle{ \mathbb{R}^{+} }[/math]. The proof proceeds as follows. Let [math]\displaystyle{ f }[/math] be a non-negative measurable function defined over the measure space [math]\displaystyle{ (X, \Sigma,\mu) }[/math]. For each [math]\displaystyle{ n\in\mathbb N }[/math], subdivide the co-domain of [math]\displaystyle{ f }[/math] into [math]\displaystyle{ 2^{2n}+1 }[/math] intervals, [math]\displaystyle{ 2^{2n} }[/math] of which have length [math]\displaystyle{ 2^{-n} }[/math]. That is, for each [math]\displaystyle{ n }[/math], define

- [math]\displaystyle{ I_{n,k}=\left[\frac{k-1}{2^n},\frac{k}{2^n}\right) }[/math] for [math]\displaystyle{ k=1,2,\ldots,2^{2n} }[/math], and [math]\displaystyle{ I_{n,2^{2n}+1}=[2^n,\infty) }[/math],

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

Now define the sets

- [math]\displaystyle{ A_{n,k}=f^{-1}(I_{n,k}) \, }[/math] for [math]\displaystyle{ k=1,2,\ldots,2^{2n}+1, }[/math]

which are measurable ([math]\displaystyle{ A_{n,k}\in \Sigma }[/math]) because [math]\displaystyle{ f }[/math] is assumed to be measurable.

Then the increasing sequence of simple functions

- [math]\displaystyle{ f_n=\sum_{k=1}^{2^{2n}+1}\frac{k-1}{2^n}{\mathbf 1}_{A_{n,k}} }[/math]

converges pointwise to [math]\displaystyle{ f }[/math] as [math]\displaystyle{ n\to\infty }[/math]. Note that, when [math]\displaystyle{ f }[/math] is bounded, the convergence is uniform.

## 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.

Original source: https://en.wikipedia.org/wiki/Simple function.
Read more |