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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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 $\mu(\cup\cap E_k)=0$ really signify?? There's a pretty simple explanation if $(X,\Sigma,\mu)$ is a probability space, but how are we to understand the result in the context of general measure spaces?

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Open Sets Are Everything

In today's post I want to emphasize a simple - but important - idea in topology which I think is helpful for anyone new to the subject, and that is: Open sets are everything! What do I mean by that? Well, for a given set $X$, all the properties of $X$ are HIGHLY dependent on how you define an "open set."

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On Connectedness, Intuitively

Today's post is a bit of a ramble, but my goal is to uncover the intuition behind one of the definitions of a connected topological space. Ideally, this is just a little tidbit I'd like to stash in The Back Pocket. But as you can tell already, the length of this post isn't so "little"! Oh well, here we go!

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What is Galois Theory Anyway?

Perhaps you've heard of Évariste Galois? (Pronounced "GAL-wah.") You know, the French mathematician who died tragically in 1832 in a duel at the tender age of 20? (Supposedly over a girl! C'est romantique, n'est-ce pas?)  Well, today we're taking a bird's-eye view of his most well-known contribution to mathematics: the appropriately named Galois theory. The goal of this post is twofold...

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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.

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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.)

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Why are Noetherian Rings Special?

In short, "Noetherian-ness" is a property which generalizes "PID-ness." As Keith Conrad so nicely puts it, "The property of all ideals being singly generated is often not preserved under common ring-theoretic constructions (e.g. $\mathbb{Z}$ is a PID but $\mathbb{Z}[x]$ is not), but the property of all ideals being finitely generated does remain valid under many constructions of new rings from old rings. For example... every quadratic ring $\mathbb{Z}[\sqrt{d}]$ is Noetherian, even though many of these rings are not PIDs." (italics added)

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Motivation for the Tensor Product

In general, if $F$ is a field and $V$ is a vector space over $F$, the tensor product answers the question "How can I define scalar multiplication on $V$ by some larger field which contains $F$?" (Of course this holds if we replace the word "field" by "ring" and consider the same scenario with modules.)

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One Unspoken Rule of Algebra

Here's an algebra tip! Whenever you're asked to prove $$A/B\cong C$$ where $A,B,C$ are groups, rings, fields, modules, etc., mostly likely the The First Isomorphism Theorem involved!

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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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What do Polygons and Galois Theory Have in Common?

Galois Theory is all about symmetry. So, perhaps not surprisingly, symmetries found among the roots of polynomials (via Galois theory) are closely related to symmetries of polygons in the plane (via geometry). In fact, the two are highly analogous! 

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Borel-Cantelli Lemma (Pictorially)

The Borel-Cantelli Lemma says that if $(X,\Sigma,\mu)$ is a measure space with $\mu(X)<\infty$ and if $\{E_n\}_{n=1}^\infty$ is a sequence of measurable sets such that $\sum_n\mu(E_n)<\infty$, then $$\mu\left(\bigcap_{n=1}^\infty \bigcup_{k=n}^\infty E_k\right)=\mu\left(\limsup_{n\to\infty} En \right)=0.$$

(For the record, I didn't understand this when I first saw it (or for a long time afterwards). My only thought was, "But what does that mean? In English??") To help our intuition…

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Operator Norm, Intuitively

If $X$ and $Y$ are normed vector spaces, a linear map $T:X\to Y$ is said to be <b>bounded</b> if $\|T\|< \infty$ where

$$\|T\|=\sup_{\underset{x\neq 0}{x\in X}}\left\{\frac{|T(x)|}{|x|}\right\}.$$

(Note that $|T(x)|$ is the norm in $Y$ whereas $|x|$ is the norm in $X$.) One can show that this is equivalent to

$$\|T\|=\sup_{x\in X}\{|T(x)|:|x|=1\}.$$ So intuitively (at least in two dimensions), we can think of $\|T\|$ this way…

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Need to Prove Your Ring is NOT a UFD?

You're given a ring $R$ and are asked to show it's not a UFD. Where do you begin? One standard trick is to apply the Rational Roots Theorem….

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Two Ways to be Small

  In real analysis, there are two ways a measurable set $E$ can be small. Either

  • the measure of $E$ is 0, OR
  • $E$ is nowhere dense.

Intuitively, to say the measure of $E$ is $0$ means that...

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One Unspoken Rule of Measure Theory

Here's a measure theory trick: when asked to prove that a set of points in $\mathbb{R}$ (or some measure space $X$) has a certain property, try to show that the set of points which does NOT have that property has measure 0! This technique is used quite often.

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What's a Transitive Group Action?

Let a group $G$ act on a set $X$. The action is said to be transitive if for any two $x,y\in X$ there is a $g\in G$ such that $g\cdot x=y$. This is equivalent to saying there is an $x\in X$ such that $\text{orb}(x)=X$, i.e. there is exactly one orbit. And all this is just the fancy way of saying that $G$ shuffles all the elements of $X$ among themselves. In other words…

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Completing a Metric Space, Intuitively

An incomplete metric space is very much like a golf course: it has a lot of missing points!

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Baire Category & Nowhere Differentiable Functions (Part One)

The Baire Category Theorem is a powerful result that relates a metric space to its underlying topology. (And sadly no, nothing to do with category theory!) Informally, the theorem says that if you can find a metric with respect to which your topological space is complete, then that space cannot be written as a countable union of nowhere dense sets. In other words, a metric can put a restriction on the topology.

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Continuous Functions, Discontinuous Supremum

A function $f$ is said to be continuous if the preimage of any open set is open. Analogously, we might say that a function is measurable if the preimage of a measurable set is measurable. It's not hard to show that if $\{f_n\}$ is a sequence of measurable functions, then sup$\{f_n\}$, inf$\{f_n\}$, limsup $\{f_n\}$ and liminf $\{f_n\}$ are also measurable functions. But here the analogy between continuity and measurability breaks down. It is not true that if each $f_n$ is a continuous function, then sup$\{f_n\}$, inf$\{f_n\}$, limsup $\{f_n\}$ and liminf $\{f_n\}$ are continuous as well. Below is a counterexample - a sequence of continuous functions with a discontinuous supremum!

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Need Some Disjoint Sets? (A Measure Theory Trick)

Given a countable collection of measurable sets, is it possible to construct a new collection of sets which are pairwise disjoint and have the same union as the original? Yes! Here's the trick....

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On Constructing Functions, Part 6

This post is the sixth example in an ongoing list of various sequences of functions which converge to different things in different ways. Today we have a sequence of functions on $[0,1]$ which converges to 0 in $L^1$, but does not converge anywhere on $[0,1]$.

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Stone Weierstrass Theorem (Example)

This week we continue our discussion on the Stone Weierstrass Theorem with an example. This exercise is taken from Rudin's Principles of Mathematical Analysis (affectionately known as "Baby Rudin"), chapter 7 #20.

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Stone Weierstrass Theorem

The Stone Weierstrass Theorem is a generalization of the familiar Weierstrass Approximation Theorem. In this post, we introduce the Stone Weierstrass Theorem and, by looking at counterexamples, discover why each of the hypotheses of the theorem are necessary.

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Rational Canonical Form: Example #2 (with Galois Theory)

Last week we saw an example of how to use the rational canonical form (RCF) to classify matrices of a given order in $GL_2(\mathbb{Q})$. Today we have a similar example (taken from CUNY's spring 2015 qualifying exam) where now our matrices have entires in the finite field $F_13$. The fact that our field is $F_13$ instead of $\mathbb{Q}$ actually makes little difference in how to approach the solution, but I think this problem is particularly nice because part of it calls on some Galois Theory.

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Rational Canonical Form: Example #1

Last time we discussed the rational canonical form (RCF) of a linear transformation, and we mentioned that any two similar linear transformations have the same RCF. It's this fact which allows us to classify distinct linear transformations on a given $F$-vector space $V$ for some field $F$. Today, to illustrate this, we'll work through a concrete example:

Find representatives for the distinct conjugacy classes of matrices of finite order in the multiplicative group of 2x2 matrices with rational entries.

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Rational Canonical Form: A Summary

This post is intended to be a hopefully-not-too-intimidating summary of the rational canonical form (RCF) of a linear transformation. Of course, anything which involves the word "canonical" is probably intimidating no matter what. But even so, I've attempted to write a distilled version of the material found in (the first half of) section 12.2 from Dummit and Foote's Abstract Algebra.

In sum, the RCF is important because it allows us to classify linear transformations on a vector space up to conjugation. Below we'll set up some background, then define the rational canonical form, and close by discussing why the RCF looks the way it does. Next week we'll go through an explicit example to see exactly how the RCF can be used to classify linear transformations.

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Finitely Generated Modules Over a PID

We know what it means to have a module $M$ over a (commutative, say) ring $R$. We also know that if our ring $R$ is actually a field, our module becomes a vector space. But what happens if $R$ is "merely" a PID? Answer: A lot. Today we'll look at a proposition, which, thanks to the language of exact sequences, is quite simple and from which the Fundamental Theorem of Finitely Generated Modules over a PID follows almost immediately. The information below is loosely based on section 12.1 of Dummit and Foote' Abstract Algebra.

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A Little Fact From Group Actions

Today we've got a little post on a little fact relating to group actions. I wanted to write about this not so much to emphasize its importance (it's certainly not a major result) but simply to uncover the intuition behind it.

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Two Tricks Using Eisenstein's Criterion

Today we're talking about Eisenstein's (not Einstein's!) Criterion - a well-known test to determine whether or not a polynomial is irreducible. In particular, we'll consider two examples where a not-so-obvious trick is needed in order to apply the criterion.

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Noetherian Rings = Generalization of PIDs

When I was first introduced to Noetherian rings, I didn't understand why my professor made such a big hoopla over these things. What makes Noetherian rings so special? Today's post is just a little intuition to stash in The Back Pocket, for anyone hearing the word "Noetherian" for the first time. 

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4 Ways to Show a Group is Not Simple

You know the Sylow game. You're given a group of a certain order and are asked to show it's not simple. But where do you start? Here are four options that may be helpful when trying to produce a nontrivial normal subgroup.

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On Constructing Functions, Part 5

This post is the fifth example in an ongoing list of various sequences of functions which converge to different things in different ways. Today we have a sequence of functions which converges to 0 pointwise but does not converge to 0 in $L^1$.

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Maximal ≠ Maximum!

Suffixes are important! 

Did you know that the words

"maximal" and "maximum" generally do NOT mean the same thing

in mathematics? It wasn't until I had to think about Zorn's Lemma in the context of maximal ideals that I actually thought about this, but more on that in a moment. Let's start by comparing the definitions:

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On Constructing Functions, Part 4

This post is the fourth example in an ongoing list of various sequences of functions which converge to different things in different ways. Here's a sequence of Lebesgue integrable functions which converges uniformly to a function which is not Lebesgue integrable.

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Algebraic Elements Are Like Limit Points!

When you hear the word closure, what do you think of? I think of wholeness - you know, tying loose ends, wrapping things up, filling in the missing parts. This same idea is behind the mathematician's notion of closure, as in the phrase "taking the closure" of a set. Intuitively this just means adding in any missing pieces so that the result is complete, whole.

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Constructing the Tensor Product of Modules

Today we talk tensor products. Specifically this post covers the construction of the tensor product between two modules over a ring. But before jumping in, I think now's a good time to ask, "What are tensor products good for?" Here's a simple example where such a question might arise...

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On Constructing Functions, Part 3

This post is the third example in an ongoing list of various sequences of functions which converge to different things in different ways. Here's a sequence of continuous functions which converges in the $L^1$ norm (the set of Lebesgue measurable functions), but does not converge uniformly.

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On Constructing Functions, Part 2

This post is the second example in an ongoing list of various sequences of functions which converge to different things in different ways. Here's a sequence which converges uniformly but does not converge in $L^1$ (the set of Lebesgue measurable functions).

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On Constructing Functions, Part 1

Given a sequence of real-valued functions $\{f_n\}$, the phrase, "$f_n$ converges to a function $f$" can mean a few things:

  • $f_n$ converges uniformly
  • $f_n$ converges pointwise
  • $f_n$ converges almost everywhere (a.e.)
  • $f_n$ converges in $L^1$ (set of Lebesgue integrable functions)
  • and so on...

Other factors come into play if the $f_n$ are required to be continuous, defined on a compact set, integrable, etc.. So since I do not have the memory of an elephant (whatever that phrase means...), I've decided to keep a list of different sequences that converge (or don't converge) to different functions in different ways. With each example I'll also include a little (and hopefully) intuitive explanation for why. Having these sequences close at hand is  especially useful when analyzing the behavior of certain functions or constructing counterexamples.

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The Integral Domain Hierarchy, Part 2

In any area of math, it's always good idea to keep a few counterexamples in your back pocket. This post continues part 1 with examples/non-examples from some of the different subsets of integral domains. 

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The Integral Domain Hierarchy, Part 1

Here is a list of some of the subsets of integral domains, along with the reasoning (a.k.a proofs) of why the bullseye below looks the way it does. Part 2 of this post will include back-pocket examples/non-examples of each.   

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Compact + Hausdorff = Normal

The notion of a topological space being Hausdorff or normal identifies the degree to which points or sets can be "separated." In a Hausdorff space, it's guaranteed that if you pick any two distinct points in the space -- say $x$ and $y$ -- then you can always find an open set containing $x$ and an open set containing $y$ such that those two sets don't overlap.

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Ways to Show a Group is Abelian

After some exposure to group theory, you quickly learn that when trying to prove a group $G$ is abelian, checking if $xy=yx$ for arbitrary $x,y$ in $G$ is not always the most efficient - or helpful! - tactic. Here is a (not comprehensive) running tab of other ways you may be able to prove your group is abelian:

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A Math Blog? Say What?

Yes! I'm writing about math. No! Don't close your browser window. Hear me out first...

I know very well that math has a bad rap. It's often taught or thought of as a dry, intimidating, unapproachable, completely boring, who-in-their-right-mind-would-want-to-think-about-this-on-purpose kind of subject. I get it. Math was the last thing on earth I thought I'd study. Seriously.

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