Measure Theory for Computer Scientists
In recent years, measure theory has been coming up in my research a few times. As a computer science Ph.D. with a focus on optimization and AI methods, this was a mathematical tool that I had not had any exposure to. In this post I give an overview of the sources that I found most helpful (and enjoyable!) for learning measure theory.
Getting Started
Because measure theory is pretty abstract, it can be a bit boring if started without some appreciation for why it is necessary. To that end, I recommend starting with some lighter and more fun approaches to measure theory. I found the below two sources very enjoyable.

Mathematics++: Selected Topics Beyond the Basic Courses by Kantor, Matoušek, and Šámal. Chapter 1 of this book provide a really nice and enjoyable introduction to measure theory. Lots of little exercises are interspersed, ensuring comprehension. Finally, the book is explicitly written with an eye towards doctoral students in computer science, so it perfectly fits the background I had.

The Bright Side of Mathematics Lectures Very nice videos for complementing measure theory readings. Each video is about 1025 minutes long, so they can easily be watched as a short distraction whenever convenient.
A Bit More Depth
 Measures, Integrals and Martingales by Rene Schilling. A bit more approachable than Rosenthal’s book.
 A First Look at Rigorous Probability Theory by Jeffrey S. Rosenthal.
Measure Theory for Economics Research
For me, measure theory was largely coming up in the context of proving the existence of equilibria in infinitedimensional games and markets. Below are some resources specifically helpful for that. These are generally not gentle, and would be best used either as reference works, or after studying some of the above resources first.

Infinite dimensional analysis: A Hitchhiker’s Guide by Aliprantis and Border. Quoting from the preface: ``We realized that the typical graduate student in mathematical economics has to be familiar with a vast amount of material that spans several traditional fields in mathematics. Much of the material appears only in esoteric research monographs that are designed for specialists, not for the sort of generalist that our students need be.’’ This book, which is epic in scope, fixes this problem. This book has very nice coverage of measure theory, but also a lot of useful content on functional analysis, convex analysis, fixedpoint theorems, and so on.

Wikipedia Good for a quick overview of several of the infinitedimensional fixedpoint theorems