---
tag: 0565EGK6XP
kind: statement
role: theorem
name: Lebesgue's criterion for Riemann integrability
aliases: [lebesgue-vitali-theorem]
msc: [26A42]
uses: [0N1JQXZXHC, 0EEE2CZB2A]
sources:
  - url: https://en.wikipedia.org/wiki/Riemann_integral
    locator: Integrability
---

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded.
Then $f$ is [[Riemann integrable|0N1JQXZXHC]] if and only if the set of points where $f$ is not [[continuous|0EEE2CZB2A]] has Lebesgue measure zero.

## Remarks

A set $E \subseteq \mathbb{R}$ has measure zero if for each $\varepsilon > 0$ it is covered by countably many open intervals of total length less than $\varepsilon$.
Countable sets have measure zero. So [[Thomae's function|0HJ28H9Y95]] is integrable, and so is each monotone function.
The [[Dirichlet function|03Z4JC4YRJ]] is discontinuous everywhere, so it is not integrable.
Giuseppe Vitali and Henri Lebesgue proved the criterion independently, in 1907.

## Explanations

### L1 human

A bounded function is Riemann integrable exactly when it is continuous at almost all points.
The bad points must form a set of total length zero.