---
tag: 0R0T1VWH5X
kind: statement
role: theorem
name: Fubini's theorem
aliases: [fubini-theorem]
msc: [26B15]
uses: [0NKBMFC18R, 0P4WX7Y502, 0S9EK5MQPE]
sources:
  - url: https://en.wikipedia.org/wiki/Fubini%27s_theorem
    locator: Riemann integrals
  - url: https://en.wikipedia.org/wiki/Fubini%27s_theorem
    locator: For integrable functions
  - url: https://en.wikipedia.org/wiki/Fubini%27s_theorem
    locator: History
same_as:
  wikidata: Q1149022
  wikipedia: Fubini's theorem
---

## Statement

Let $R = [a, b] \times [c, d]$, and let $f \colon R \to \mathbb{R}$ be Riemann integrable on $R$.
Suppose that $y \mapsto f(x, y)$ is integrable on $[c, d]$ for each $x \in [a, b]$, and that $x \mapsto f(x, y)$ is integrable on $[a, b]$ for each $y \in [c, d]$.
Then
$$\iint_R f \, dA = \int_a^b \int_c^d f(x, y) \, dy \, dx = \int_c^d \int_a^b f(x, y) \, dx \, dy.$$
In particular, this holds for each [[continuous|0S9EK5MQPE]] function $f$ on $R$.

## Remarks

The left side is the [[double integral|0NKBMFC18R]], and the other two are [[iterated integrals|0P4WX7Y502]].
The Lebesgue version: let $f$ be Lebesgue measurable on $\mathbb{R}^n \times \mathbb{R}^m$ with $\int |f| < \infty$. Then the double integral equals both iterated integrals, where the inner integrals exist for almost all values of the outer variable.
Without integrability of $|f|$, the [[iterated integrals can differ|0GJZE2E4RX]]. For non-negative functions, [[Tonelli's theorem|067SYD2J4B]] removes the integrability hypothesis.
Euler knew the case of continuous functions. Lebesgue proved a version for bounded measurable functions in 1904, and Fubini proved the version for integrable functions in 1907.

## Explanations

### L1 human

To integrate over a rectangle, you can slice it into lines, integrate along each line, and then integrate the results.
Fubini's theorem says that both slicing directions give the same answer as the double integral, for well-behaved functions.