## Analysis

# Automorphisms of the Riemann Sphere

This is the last in a four-part series in which we prove that the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere each take on a different form. Today our focus is on the Riemann sphere.

# Automorphisms of the Complex Plane

This is part three of a four-part series in which we prove that the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere each take on a different form. Today our focus is on the complex plane.

# Automorphisms of the Upper Half Plane

This is part two of a four-part series in which we prove that the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere each take on a different form. Today our focus is on the upper half plane.

# Automorphisms of the Unit Disc

This is part one of a four-part series in which we prove that the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere each take on a different form. Today our focus is on the unit disc.

# Three Important Riemann Surfaces

In this post we ramble on about Riemann surfaces, the uniformization theorem, universal covers, and two secret (or not-so-secret!) techniques that mathematicians use to study a given space. Our intent is to provide motivation for an upcoming mini-series on the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere.

# The Pseudo-Hyperbolic Metric and Lindelöf's Inequality (cont.)

Last time we proved that the pseudo-hyperbolic metric on the unit disc in ℂ is indeed a metric. In today’s post, we use this fact to verify Lindelöf’s inequality which says, "Hey! Want to apply Schwarz's Lemma but don't know if your function fixes the origin? Here's what you *do *know...."

# The Pseudo-Hyperbolic Metric and Lindelöf's Inequality

In this post, we define the pseudo-hyperbolic metric on the unit disc in ℂ and prove it does indeed satisfy the conditions of a metric.

# 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 $\mu(\cup\cap E_k)=0$ 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?

# 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.)

# 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.

# 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.

# 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!

# Need Some Disjoint Sets? (A Measure Theory Trick)

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

# 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 $[0,1]$.

# 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.

# 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.

# 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 $L^1$.

# 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.

# 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.

# 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).

# 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.