---
tag: 7WMBX7M4P6
kind: concept
name: Simple function
msc: [28A20]
specializes: [YYNCJSR0PJ]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_function
same_as:
  wikidata: Q913022
  wikipedia: Simple function
---

## Summary

A simple function is a [[measurable function|YYNCJSR0PJ]] that takes only finitely many values.
It is a finite combination of [[indicator functions|0RCQZ36MME]] of measurable sets, so its integral is a finite sum.
The Lebesgue integral of a nonnegative function is defined by approximating it from below by simple functions.

## Explanations

### L1 human

A simple function is like a staircase, except that each step may sit on a complicated set instead of an interval.
On each set it has a constant value, and there are only finitely many sets.
For example, the function on $[0, 1]$ that is $2$ on the rational numbers and $5$ on the irrational numbers is simple.
Its integral is $2$ times the measure of the rationals plus $5$ times the measure of the irrationals, which is $0 + 5 = 5$.