---
title: "Measurable Functions and Approximation by Simple Functions"
description: "A function is measurable when the preimage of every interval is a measurable set. Every non-negative measurable function is an increasing limit of simple functions, and that staircase is what makes an"
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retrieved: 2026-09-12
---

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# Measurable Functions and Approximation by Simple Functions

A function is measurable when the preimage of every interval is a measurable set. Every non-negative measurable function is an increasing limit of simple functions, and that staircase is what makes an integral definable in stages.

Subject: Mathematics · Area: Calculus & Analysis · Ages 22 to 23
Page: https://lightmysky.com/learn/mathematics/measurable-functions-and-approximation-by-simple-functions-mt_VwNRjXA7ro

## Ready when they can

- Test measurability using preimages of open rays
- Build an increasing sequence of simple functions converging to a given non-negative function
- Show that a pointwise limit of measurable functions stays measurable, which fails for continuous functions

## Lesson: Functions that respect measurable sets

A function is measurable when the preimage of every interval is a measurable set. In practice you only check open rays, the sets of points where the value exceeds a number a. If each of those preimages is measurable, the function counts as measurable.

**Example.** A simple function takes only finitely many values, one on each piece of a measurable partition. To approximate a non-negative function f, slice its range into thin layers and build a staircase that stays just below f. Make the layers thinner and the staircase climbs up to f, step by step.

A pointwise limit of measurable functions stays measurable. The reason is that the limit can be rewritten with countably many unions and intersections of preimages, and measurable sets survive those operations. So the limit function passes the preimage test too.

**Tip.** Do not mix this up with continuous functions. A pointwise limit of continuous functions can be discontinuous, like powers x to the n on [0, 1] climbing to a jump at 1. Measurable functions survive pointwise limits, while continuous ones do not.

**Recap.** Measurable functions pull intervals back to measurable sets, which is why staircases can approximate them and pointwise limits cannot escape the class.

## Practice

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

## Needs first

- [Outer Measure and the Construction of Lebesgue Measure](https://lightmysky.com/learn/mathematics/outer-measure-and-the-construction-of-lebesgue-measure-mt_6u5_Um71wE)
- [Pointwise and Uniform Convergence](https://lightmysky.com/learn/mathematics/pointwise-and-uniform-convergence-mt_iAGk1ML5MA)

## Opens up

- [The Lebesgue Integral and What It Repairs](https://lightmysky.com/learn/mathematics/the-lebesgue-integral-and-what-it-repairs-mt_KO92nLc5YU)