# Commutative Diagrams Explained

Have you ever come across the words "commutative diagram" before? Perhaps you've read or heard someone utter a sentence that went something like

"For every [bla bla] there exists
the following diagram commutes."

and perhaps it left you wondering what it all meant.

# The Most Obvious Secret in Mathematics

Yes, I agree. The title for this post is a little pretentious. It's certainly possible that there are other mathematical secrets that are more obvious than this one, but hey, I got your attention, right? Good. Because I'd like to tell you about an overarching theme in mathematics - a mathematical mantra, if you will. A technique that mathematicians use all the time to, well, do math.

# Resources for Intro-Level Graduate Courses

In recent months, several of you have asked me to recommend resources for various subjects in mathematics. Well, folks, here it is! I've finally rounded up a collection of books, PDFs, videos, and websites that I found helpful while studying for my intro-level graduate courses.

# Good Reads: Real Analysis by N. L. Carothers

Have you been on the hunt for a good introductory-level real analysis book? Look no further! The underrated Real Analysis by N. L. Carothers is, in my opinion, one of the best out there. Real analysis has a reputation for being a fearful subject for many students, but this text by Carothers does a great job of mitigating those fears. Aimed towards advanced undergraduate and early graduate students, it is written in a fantastically warm and approachable manner without sacrificing too much rigor. The text is intentionally conversational (which I love!) and includes plenty of exercises and illustrations, all the while informing the reader of context and historical background along the way.

# Absolute Continuity (Part Two)

There are two definitions of absolute continuity out there. One refers to an absolutely continuous function and the other to an absolutely continuous measure. And although the definitions appear unrelated, they are in fact very much related, linked together by Lebesgue's Fundamental Theorem of Calculus. This is the second of a two-part series where we explore that relationship.

# Absolute Continuity (Part One)

There are two definitions of absolute continuity out there. One refers to an absolutely continuous function and the other to an absolutely continuous measure. And although the definitions appear unrelated, they are in fact very much related, linked together by Lebesgue's Fundamental Theorem of Calculus. This is part one of a two-part series where we explore that relationship.

# Dominated Convergence Theorem

Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem are three major results in the theory of Lebesgue integration which, when given a sequence of functions {f_n}, answer the question, "When can I switch the limit symbol and the integral symbol?" In this post, we discuss the Dominated Convergence Theorem and see why "domination" is necessary.

# Monotone Convergence Theorem

Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem are three major results in the theory of Lebesgue integration which, when given a sequence of functions {f_n}, answer the question, "When can I switch the limit symbol and the integral symbol?" In this post, we discuss the Monotone Convergence Theorem and solve a nasty-looking problem which, thanks to the theorem, is actually quite trivial!

# Fatou's Lemma

Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem are three major results in the theory of Lebesgue integration which, when given a sequence of functions {f_n}, answer the question, "When can I switch the limit symbol and the integral symbol?" In this post, we discuss Fatou's Lemma and solve a problem from Rudin's Real and Complex Analysis (a.k.a. "Big Rudin").

# The Borel-Cantelli Lemma

Today we're chatting about the Borel-Cantelli Lemma. When I first came across this lemma, I struggled to understand what it meant "in English." What does μ(∩∪E_k)=0 really signify??  There's a pretty simple explanation if (X,Σ,μ) is a probability space, but how are we to understand the result in the context of general measure spaces?

# Baire Category & Nowhere Differentiable Functions (Part Two)

Welcome to part two of our discussion on Baire's Category Theorem. Today we'll sketch the proof that we can find a continuous function on [0,1] which is nowhere differentiable.

# A Non-Measurable Set

Today we're looking at a fairly simple proof of a standard result in measure theory:

Theorem: Any measurable subset A of the real line with positive measure contains a non-measurable subset.

(Remark: we used this theorem last week to prove the existence of a Lebesgue measurable set which is not a Borel set.)