Layman's Introduction to Measure Theory

Measure theory studies ways of generalizing the notions of length/area/volume. Even in 2 dimensions, it might not be clear how to measure the area of the following fairly tame shape:

What's the area of this shape?

much less the “area” of even weirder shapes in higher dimensions or different spaces entirely.

For example, suppose you want to measure the length of a book (so that you can get a good sense of how long it takes to read). What’s a good measure? One possibility is to measure a book’s length in pages. Since books provide page counts, this is a fairly easy measure to get. However, different versions of the same book (e.g., hardcover and paperback versions) tend to have different page counts, so this page measure doesn’t satisfy the nice property of version invariance (which we would like to have, since hardcover and paperback versions of the same book take the same time to read). Also, not all books even have page counts (think Kindle books), so this measure doesn’t allow us to measure the length of all books we might want to read.

Another, possibly better measure is to measure a book’s length in terms of the number of words it contains. Now we do have version invariance (hardcover and paperback versions contain the same number of words) and we can measure the length of Kindle books as well. We can even do things like add two books together, and the measure/number of words of the concatenated books will pleasantly equal the sum of the measures/number of words of each book alone.

However, what happens when we try to measure a picture book’s length in words? We can’t – picture books are too pathological. Maybe we could say that a picture book has measure zero (since a picture book has no words), but then we get unhappy things like books of measure zero taking a really long time to read (imagine a really long picture book). So maybe a better option is to say that picture books are simply unmeasurable. Whenever someone asks for the length of a picture book, we ignore them, and this way our measure will continue to be a good approximation of reading time and we get to keep our other nice properties as well.

Similarly, measure theory asks questions like:

  • How do we define a measure on our space? (Jordan measure and Lebesgue measure are two different options in Euclidean space.)
  • What properties does our measure satisfy? (For example, does it satisfy translational invariance, rotational invariance, additivity?)
  • Which objects are measurable/which objects can we say it’s okay not to measure in order to preserve nice properties of our measure? (The Banach-Tarski ball can be rigidly reassembled into two copies of the same shape and size as the original, so we don’t want it to be measurable, since then we would lose additivity properties.)

And once we’ve defined a “generalized area” (our measure), we can try to generalize other mathematical concepts as well. For example, recall that the (Riemann) integral that you learn in calculus measures the area under a curve. What happens if we replace the “area” in the Riemann integral with our new, generalized measure (e.g., to get the Lebesgue integral)? Measure theory also helps make certain probability statements mathematically precise (e.g., we can say exactly what it means that a fair coin flipped infinitely often will “almost never” land heads more than 50% of the time).

Edwin Chen

Founder at Surge AI, the world's most powerful data labeling platform and workforce for NLP.

Need obsessively high-quality human-powered data? Reach out! We help top AI companies like OpenAI, Amazon, and Airbnb create stunning high-skill, human-labeled datasets.

Former AI & engineering lead at Google, Facebook, Twitter, Dropbox, and MSR. Pure math, theoretical CS, and linguistics at MIT.

Surge AI
Surge AI Twitter
Surge AI Blog
Surge AI Github
Surge AI LinkedIn


Recent Posts

How Could Facebook Align its ML Systems to Human Values? A Data-Driven Approach

A Visual Tool for Exploring Word Embeddings

A Layman's Introduction to Perplexity in NLP

An Introduction to Inter-Annotator Agreement and Cohen's Kappa Statistic

A Visual, Layman's Introduction to Language Models in NLP

Surge AI: A New Data Labeling Platform and Workforce for NLP

Exploring LSTMs

Moving Beyond CTR: Better Recommendations Through Human Evaluation

Propensity Modeling, Causal Inference, and Discovering Drivers of Growth

Product Insights for Airbnb

Improving Twitter Search with Real-Time Human Computation

Edge Prediction in a Social Graph: My Solution to Facebook's User Recommendation Contest on Kaggle

Soda vs. Pop with Twitter

Infinite Mixture Models with Nonparametric Bayes and the Dirichlet Process

Instant Interactive Visualization with d3 + ggplot2

Movie Recommendations and More via MapReduce and Scalding

Quick Introduction to ggplot2

Introduction to Conditional Random Fields

Winning the Netflix Prize: A Summary

Stuff Harvard People Like

Information Transmission in a Social Network: Dissecting the Spread of a Quora Post

Introduction to Latent Dirichlet Allocation

Introduction to Restricted Boltzmann Machines

Topic Modeling the Sarah Palin Emails

Filtering for English Tweets: Unsupervised Language Detection on Twitter

Choosing a Machine Learning Classifier

Kickstarter Data Analysis: Success and Pricing

A Mathematical Introduction to Least Angle Regression

Introduction to Cointegration and Pairs Trading

Counting Clusters

Hacker News Analysis

Layman's Introduction to Measure Theory

Layman's Introduction to Random Forests

Netflix Prize Summary: Factorization Meets the Neighborhood

Netflix Prize Summary: Scalable Collaborative Filtering with Jointly Derived Neighborhood Interpolation Weights

Prime Numbers and the Riemann Zeta Function

Topological Combinatorics and the Evasiveness Conjecture

Item-to-Item Collaborative Filtering with Amazon's Recommendation System