---
tag: RARTMA0ZEC
kind: concept
name: Almost everywhere
msc: [28A12, 28A20]
sources:
  - url: https://en.wikipedia.org/wiki/Almost_everywhere
same_as:
  wikidata: Q1139334
  wikipedia: Almost everywhere
---

## Summary

A property of the points of a measure space holds almost everywhere if the points where it fails lie in a set of measure zero.
Most statements about integrals and measurable functions only hold, and only need to hold, almost everywhere.
In probability theory the same notion is called almost surely.

## Explanations

### L1 human

"Almost everywhere" means "everywhere except on a negligible set".
The function that is $1$ at rational numbers and $0$ elsewhere is $0$ almost everywhere on the real line.
The rational numbers form a countable set, which has length zero.
Two functions that agree almost everywhere have the same integral, so measure theory treats them as the same.