Measure Theory for Beginners: An Intuitive Approach
For most mathematicians and statisticians, some physicists, and some other various scientists, one of the hardest subjects they encounter is measure theory. Measure theory is the theory which underlies most of modern statistics and analysis, and in general is a very subtle and deep subject. What makes it so difficult for many is this subtlety; the material is very much rooted in the more abstract set theory that arose out of the early 20th century, and doesn’t read quite like other topics in Mathematics. The way much of measure theory is taught, the theorems are dense and require a working knowledge of topology, some background in set theory w/r/t infinite cardinalities and structures, and some other various topics in analysis.
It doesn’t have to be this way! In this article, I hope to cover some basics of measure theory for those who are unfamiliar, and to offer some insight into the motivations for why and how we approach it the way we do. We will cover two sides of measure theory, the probability side and the analytic side (and why they are the same). This article is written for those with a basic level of undergraduate real analysis or advanced calculus, or other STEM majors who are willing to put a little work in to learn some definitions.
In this article, I cover up to Lebesgue measure and measurable sets. I will provide a second article in the future discussing Lebesgue integration and some more analysis discussion.
