# 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 a 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 measurable sets, and let *a*_{1}, ..., *a*_{n} be a sequence of real or complex numbers. A *simple function* is a function of the form

where 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 .

## Integration of simple functions

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

if all summands are finite.

## Relation to Lebesgue integration

Any non-negative measurable function is the pointwise limit of a monotonic increasing sequence of non-negative simple functions. Indeed, let be a non-negative measurable function defined over the measure space as before. For each , subdivide the range of into intervals, of which have length . For each , set

(Note that, for fixed , the sets are disjoint and cover the non-negative real line.)

Now define the measurable sets

Then the increasing sequence of simple functions

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

## References

- Template:Aut.
*Introduction to Measure and Probability*, 1966, Cambridge. - Template:Aut.
*Real and Functional Analysis*, 1993, Springer-Verlag. - Template:Aut.
*Real and Complex Analysis*, 1987, McGraw-Hill. - Template:Aut.
*Real Analysis*, 1968, Collier Macmillan.