A **simple function** is measurable with a finite number of real or complex values in its range (excluding infinity) [1]. Some authors omit the “measurable” constraint, defining simple functions as one that takes on a finite number of values [2]. These well-behaved functions make integration, as well as theories and proofs, much easier.

A couple of examples:

- The floor function over the half-closed interval [1, 7) only takes on the values {1, 2, 3, 4, 5, 6}.
- The real-valued Dirichlet function takes on a value of 1 if
*x*is a rational number and 0 otherwise.

A simple function has a unique representation [3] and can be written as a finite linear combination

for the real constants α

_{1}, …, α

_{n}∈ ℝ and measurable sets

*A*

_{1}, …,

*A*

_{n}∈

*S*.

The set of all real-valued simple functions is sometimes denoted as *Simp*(ℝ) [4].

Outside of real analysis, the term *simple function* is sometimes used informally to mean any function that doesn’t have complicated recurrences, summations, or other challenging to compute parts. For example, the function f(x) = x^{2} might informally be called “simple”, but that function does not meet the requirement for the formal definition because the function values are not finite.

## Step Function vs. Simple Function

Simple functions are similar to step functions, but they are more general. Step functions are defined on function intervals, which are measurable, even if they are reduced to points. Therefore, every step function is a simple function [5]. However, not every simple function is a step function. Simple functions do not *have *to be defined on an interval; the Dirichlet function is an example of such a function.

Step functions don’t really exist on some spaces, but they can be substituted with simple functions for analysis. For example, if the real line is partitioned into a countable number of intervals by a simple function, then we can formulate an approximation for the simple function as a summation of a countable number of step functions [6].

## References

[1] Coleman, M. Simple Functions. Retrieved November 10, 2021 from: https://personalpages.manchester.ac.uk/staff/mark.coleman/old/341/not5.pdf

[2] Heil, C. Integration.

[3] Zitkovic, G. Lecture 2: The Lebesgue Integral. Retrieved November 10, 2021 from:

[4] Sohrab, H. (2003). Basic Real Analysis. Springer Science & Business Media.

[5] Wachsmuth, B. (2018). 7.4. Lebesgue Integral. Example 7.4.2(a): Simple Functions. Retrieved November 10, 2021 from: https://mathcs.org/analysis/reals/integ/answers/simpfun1.html

[6] Howard, R. (2015). A Signal Theoretic Introduction to Random Processes. Wiley.

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