---
title: "Monotone Convergence, Fatou and Dominated Convergence"
description: "Three theorems that say when a limit may be moved inside an integral, each with its own price. They are the reason this integral, and not the earlier one, is the one used in probability and in partial"
canonical: https://lightmysky.com/learn/mathematics/monotone-convergence-fatou-and-dominated-convergence-mt_7n6ZDkHeBe
source: https://lightmysky.com/learn/mathematics/monotone-convergence-fatou-and-dominated-convergence-mt_7n6ZDkHeBe.md
retrieved: 2026-09-12
---

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# Monotone Convergence, Fatou and Dominated Convergence

Three theorems that say when a limit may be moved inside an integral, each with its own price. They are the reason this integral, and not the earlier one, is the one used in probability and in partial differential equations.

Subject: Mathematics · Area: Calculus & Analysis · Ages 22 to 24
Page: https://lightmysky.com/learn/mathematics/monotone-convergence-fatou-and-dominated-convergence-mt_7n6ZDkHeBe

## Ready when they can

- Apply monotone convergence and point at the step where monotonicity is used
- Give a sequence for which Fatou's inequality is strict
- Find a dominating function for a given sequence, or show that none exists

## Lesson: When limits may pass inside the integral

When non-negative functions climb steadily, f1 below f2 below f3 and so on up to f, the integrals climb to the integral of f. That is monotone convergence. The climbing, the monotonicity, is the step the proof leans on: without it the conclusion can fail.

**Example.** Let each f_n be the indicator of the interval [n, n+1]: a bump of height 1 sliding right. Every integral is 1, but the bumps leave each fixed point behind, so the limit is 0 with integral 0. Here the integral of the limit is smaller, so Fatou inequality is strict.

Dominated convergence says the swap works when one integrable function g sits above all of them: each f_n stays within g in absolute value, and f_n converges to f. Then the integrals converge to the integral of f. The sliding bumps have no such g, since any cover of all bumps has infinite integral.

**Tip.** When a limit and an integral disagree, interrogate the hypotheses. Ask whether the sequence climbs monotonically. Then ask whether one integrable function dominates them all. The failing hypothesis names the broken promise.

**Recap.** Monotone convergence needs climbing, Fatou allows a gap, and dominated convergence needs one integrable ceiling, so check the price before swapping limit and integral.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Improper Integrals and Their Convergence](https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z)
- [The Lebesgue Integral and What It Repairs](https://lightmysky.com/learn/mathematics/the-lebesgue-integral-and-what-it-repairs-mt_KO92nLc5YU)

## Opens up

- [Product Measure and Fubini's Theorem](https://lightmysky.com/learn/mathematics/product-measure-and-fubinis-theorem-mt_1VzzVmMjTB)