---
title: Measurable Functions
description: These are functions between measurable spaces that behave compatibly with the underlying sigma-algebras. Learners will understand how to define these functions, analyze their limits, and use them as the foundational building blocks for Lebesgue integration.
category: mathematics
subcategory: measure-theory
difficulty: beginner, intermediate, advanced
url: /subject/measurable-functions
---

# Measurable Functions

These are functions between measurable spaces that behave compatibly with the underlying sigma-algebras. Learners will understand how to define these functions, analyze their limits, and use them as the foundational building blocks for Lebesgue integration.

## Available Resources

3 Courses • 2 Websites

## Courses

### 1. Measure Theoretic Probability

Learn measure-theoretic probability with this course from the University of Illinois! Explore measurable functions and more.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PLLyj1Zd4UWrO6VtBSiQLsNlo9QBm30nxC

### 2. Real Analysis (Math 5210)

Learn measurable functions with Francis Ding's Real Analysis (Math 5210) course! Dive into the fundamentals and build a strong foundation.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PLLFpXNanTP9WGfbjxR5kCMXQgol4bGehz

### 3. Real Analysis (MIT 18.100A)

**Author:** Casey Rodriguez

Rigorous foundations of calculus: real numbers, convergence of sequences and series, continuity, differentiability, the Riemann integral, sequences and series of functions, uniform convergence and interchange of limits. Includes 25 lecture videos, notes, problem sets and exams, and teaches how to read and construct proofs.

**Difficulty:** Intermediate | **Price:** Free

**Link:** https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/

**Tags:** real-analysis, proofs, sequences-and-series, uniform-convergence, riemann-integral

## Websites

### 1. Sheldon Axler's Book Site

Official site for Sheldon Axler's Measure, Integration & Real Analysis (Springer Graduate Texts in Mathematics 282), with the complete book free as an open-access PDF. It develops Lebesgue measure, abstract integration, Lp spaces, Hilbert spaces, and Fourier analysis from first principles.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://measure.axler.net/

**Tags:** measure-theory, lebesgue-integration, real-analysis, functional-analysis, open-textbook

### 2. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

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