---
title: "The Lebesgue Integral and What It Repairs"
description: "Define the integral first for simple functions, then as a supremum over the simple functions underneath. Functions the Riemann theory cannot touch become integrable, and the value agrees with the Riem"
canonical: https://lightmysky.com/learn/mathematics/the-lebesgue-integral-and-what-it-repairs-mt_KO92nLc5YU
source: https://lightmysky.com/learn/mathematics/the-lebesgue-integral-and-what-it-repairs-mt_KO92nLc5YU.md
retrieved: 2026-09-12
---

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# The Lebesgue Integral and What It Repairs

Define the integral first for simple functions, then as a supremum over the simple functions underneath. Functions the Riemann theory cannot touch become integrable, and the value agrees with the Riemann integral wherever that one exists.

Subject: Mathematics · Area: Calculus & Analysis · Ages 22 to 23
Page: https://lightmysky.com/learn/mathematics/the-lebesgue-integral-and-what-it-repairs-mt_KO92nLc5YU

## Ready when they can

- Integrate a simple function directly from its defining partition
- Integrate the indicator function of the rationals and say where the Riemann definition stalls
- State how the two integrals relate on a bounded interval

## Lesson: Slicing the range instead of the domain

A simple function is constant on each piece of a finite measurable partition. Its integral is the sum over pieces of value times measure. You already know the pieces are measurable, so each term makes sense, and adding them gives the whole integral.

**Example.** The indicator of the rationals is 1 on rationals and 0 elsewhere. Its integral is 0, because the rationals have measure zero and the irrationals carry the full measure. Riemann sums stall here: every subinterval holds both kinds of points, so lower sums sit at 0 while upper sums sit at 1.

Riemann cuts the domain into subintervals, so wild oscillation inside one piece ruins the sums. Lebesgue cuts the range into levels and measures how much of the domain sits at each height. Grouping by height is what tames functions like the rationals indicator.

**Tip.** On a bounded interval, every Riemann integrable function is Lebesgue integrable, with the same value. The new integral agrees with the old one wherever the old one works, and it also handles functions the old one cannot touch.

**Recap.** The Lebesgue integral prices a function by height levels instead of domain slices, which is why it digests wilder functions yet agrees with Riemann where Riemann works.

## Practice

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

## Needs first

- [Riemann Sums and the Definite Integral as a Limit](https://lightmysky.com/learn/mathematics/riemann-sums-and-the-definite-integral-as-a-limit-mt_pwa2SRU6P9)
- [Measurable Functions and Approximation by Simple Functions](https://lightmysky.com/learn/mathematics/measurable-functions-and-approximation-by-simple-functions-mt_VwNRjXA7ro)

## Opens up

- [Monotone Convergence, Fatou and Dominated Convergence](https://lightmysky.com/learn/mathematics/monotone-convergence-fatou-and-dominated-convergence-mt_7n6ZDkHeBe)
- [Conditional Expectation Given a Sigma-Algebra](https://lightmysky.com/learn/mathematics/conditional-expectation-given-a-sigma-algebra-mt_i2sxZc-ER8)
- [Lp Spaces and the Inequalities They Rest On](https://lightmysky.com/learn/mathematics/lp-spaces-and-the-inequalities-they-rest-on-mt_Qd8jU26qgE)