---
tag: YYNCJSR0PJ
kind: concept
name: Measurable function
msc: [28A20]
sources:
  - url: https://en.wikipedia.org/wiki/Measurable_function
same_as:
  wikidata: Q516776
  wikipedia: Measurable function
---

## Summary

A measurable function is a map between [[measurable spaces|N4KDT83V3S]] under which the preimage of every measurable set is measurable.
These are the functions that one can integrate in Lebesgue's theory, and in probability they are the random variables.
Sums, products, limits and suprema of sequences of real measurable functions are again measurable.

## Explanations

### L1 human

To integrate a function, one needs to know how big the set is where the function lies in a given range.
A measurable function is one for which all these sets can be measured.
For example, on the real line, every continuous function is measurable, and so is the function that is $1$ on the rational numbers and $0$ elsewhere.
Measurable functions are the analogue for measure theory of continuous functions in topology.