Commutative Diagrams Explained

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
a [yadda yadda] such that
the following diagram commutes."

and perhaps it left you wondering what it all meant.

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The Most Obvious Secret in Mathematics

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. 

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Resources for Intro-Level Graduate Courses

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.

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Good Reads: Real Analysis by N. L. Carothers

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. 

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Absolute Continuity (Part Two)

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.

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Absolute Continuity (Part One)

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.

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Dominated Convergence Theorem

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.

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Monotone Convergence Theorem

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!

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Fatou's Lemma

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").

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Lebesgue Measurable But Not Borel

Lebesgue Measurable But Not Borel

Our goal for today is to construct a Lebesgue measurable set which is not a Borel set. In summary, we will define a homeomorphism from [0,1] to [0,2] which will map a (sub)set (of the Cantor set) of measure 0 to a set of measure 1. This set of measure 1 contains a non-measurable subset, say N. And the preimage of N will be Lebesgue measurable but will not be a Borel set. 

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