---
title: "Outer Measure and the Construction of Lebesgue Measure"
description: "Cover a set by intervals, take the cheapest cover, and every subset gets an outer measure. The Caratheodory condition then selects the sets on which that outer measure is additive, and those sets are "
canonical: https://lightmysky.com/learn/mathematics/outer-measure-and-the-construction-of-lebesgue-measure-mt_6u5_Um71wE
source: https://lightmysky.com/learn/mathematics/outer-measure-and-the-construction-of-lebesgue-measure-mt_6u5_Um71wE.md
retrieved: 2026-09-12
---

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# Outer Measure and the Construction of Lebesgue Measure

Cover a set by intervals, take the cheapest cover, and every subset gets an outer measure. The Caratheodory condition then selects the sets on which that outer measure is additive, and those sets are the measurable ones.

Subject: Mathematics · Area: Calculus & Analysis · Ages 22 to 23
Page: https://lightmysky.com/learn/mathematics/outer-measure-and-the-construction-of-lebesgue-measure-mt_6u5_Um71wE

## Ready when they can

- Compute the outer measure of a countable set and of an interval
- State the Caratheodory condition and use it to test a set for measurability
- Describe how a non-measurable set is built and which axiom the construction relies on

## Lesson: Measuring every set from the outside

You cover a set with open intervals and add up their lengths. The cheapest such total, the infimum over all covers, is the outer measure. Every subset of the line gets one. A countable set gets outer measure zero, because you can cover its points with tiny intervals whose lengths sum to almost nothing. An interval gets its usual length.

Outer measure is only subadditive: the measure of a union is at most the sum of the measures. The reason is that covers of the parts join into a cover of the whole, which gives an inequality, not an equality. Equality can fail for strange sets, so you cannot yet treat outer measure like a proper size.

**Example.** A set E is measurable when it splits every test set A cleanly: the outer measure of A equals the outer measure of the part inside E plus the outer measure of the part outside E. That equation is the Caratheodory condition. Every interval passes this test, while some far stranger sets fail it.

**Tip.** A non-measurable set is built by picking one number from each group of numbers that differ by a rational amount. That picking step uses the axiom of choice, since there are uncountably many groups and no rule names a pick. The resulting set breaks additivity, which is why it cannot be measurable.

**Recap.** Outer measure sizes every set from the outside, and the Caratheodory condition selects the sets on which that size adds up properly.

## Practice

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

## Needs first

- [Sigma-Algebras and Measurable Sets](https://lightmysky.com/learn/mathematics/sigma-algebras-and-measurable-sets-mt_cpvegazufk)
- [The Completeness Axiom: Suprema and Infima](https://lightmysky.com/learn/mathematics/the-completeness-axiom-suprema-and-infima-mt_xjI-pIfh95)

## Opens up

- [Measurable Functions and Approximation by Simple Functions](https://lightmysky.com/learn/mathematics/measurable-functions-and-approximation-by-simple-functions-mt_VwNRjXA7ro)