#royden

[spicyferretmoved]

Good Teacher Man™ Royden

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More oc refs~

[virgil-at-hot-topic]

Royce: Thanks for watching my dog.

Roman: Of course, where is he?

Cayden: Woof.

Roman: That's a person.

Royce: *petting Cayden* No.

Cayden: Woof!

[spicyferretmoved]

How does Royden feel about light puns?

At first he thought they were kinda funny, but as students kept saying them it got annoying Really Fast

[formalityisdead]

Brett Royden / switch fs flip by Marcello Guardigli Via Flickr: Brett Royden / switch fs flip - Ballarat - Australia Published in the Skateboarders Journal's web interview theskateboardersjournal.com

[brokeneyebooks]

Announcing Royden Poole's Field Guide to the 25th Hour from Clinton J. Boomer. The author returns to the world of the 25th Hour with this collection of short stories.

Have you already read The Hole Behind Midnight? Then this is the same lovable, terrible Royden you remember. Are you new to Clinton J. Boomer? Then Royden Poole's Field Guide to the 25th Hour is the perfect place to start. Pre-order today in print or ebook. It goes live April 1.

[quanticloop]

Crash course in measure theory. Part I.

Objectives

The concept of a measure and more specifically the Lebesgue measure.

Convergence of sequences of functions.

Important names: Carathéodory, Cauchy, Banach, Lebesgue, Hardy, Rudin, Borel.

Note: Royden's book is about real analysis. However, most of the theory can be extended to complex functions with very little work. (Basically, a few complex conjugaisons here and there.)

Definitions

For a thorough introduction to set theory and logical notation, on which all of this depends, I recommend this book. 

$\mathbb{R}$: The real number are an ordered field. (So are their subset $\mathbb{Q}$, the rationals.)

Upper bound of $S\subset\mathbb{R}$: $b$ is an upper bound of $S$ if $b\ge x \ \forall x \in S$. 

Lower bound of $S\subset\mathbb{R}$: $b$ is a lower bound of $S$ if $b\le x\  \forall x \in S$. 

One is of course interested in the least upper bound and the greatest lower bound for a given set. The Completeness Axiom states that every non-empty subset of $\mathbb{R}$ that has an upper bound has a least upper bound. If it has a lower bound, it has a greatest lower bound.

Extended real numbers: number system that includes $\pm \infty$ such that $\infty-\infty$ is undefined, $0 \cdot \infty =0$ and all the other binary operations are defined in the natural way.

Cauchy Criterion: a sequence of real numbers converges if and only if given $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $|x_n-x_m|<\epsilon \  \forall n,m>N$.

Cluster point of a sequence: the limit of a subsequence.

Limit superior: $\text{lim sup} \ x_n=\inf_n \sup _{k \ge n}x_k$.

Limit inferior: $\text{lim inf} \ x_n=\sup_n \inf _{k \ge n}x_k$.

Point of closure of a set: $x$ is a point of closure of $E$ if every open interval containing $x$ also contains a point of $E$.

A set is dense in $\mathbb{R}$ if its closure is $\mathbb{R}$.

Topology: The set of open subset of a space.

The collection $\mathcal{B}$ of Borel sets is the smallest $\sigma$-algebra which contains all of the open sets.

Convergences:

Pointwise: A sequence $\{f_n\}$ of functions defined on a set $E$ is said to converge pointwise on $E$ to a function $f$ if for every $x \in E$ we have $f(x)=\lim f_n(x)$. That is, if, given $x \in E$, and $\epsilon >0$, there exists $N(=N(x,\epsilon))$ such that $\forall n\ge N$, we have $|f(x)-f_n(x)|<\epsilon$.

Uniform: A sequence $\{f_n\}$ of functions defined on a set $E$ is said to converge uniformly on $E$ if, given  $\epsilon >0$ there is $N(=N(\epsilon)$ such that $\forall n\ge N$, and $ \forall x\in E$, we have $|f(x)-f_n(x)|<\epsilon$.

Lebesgue Measure on $\mathbb{R}$

The length of an interval $I$ is $l(I)$ the difference between its endpoints. It is a set function. A measure is a generalisation of this. $m: \mathcal{M} \to \mathbb{R}_{\ge 0}$ where $\mathcal{M}$ is a $\sigma$-algebra: 

$\forall E \in \mathcal{M}$, $m(E)$ is defined.

$m(I)-l(I)$ for any interval. 

If $\{E_n\}\subset \mathcal{M}$ is a sequence of disjoint sets, $m(\cup E_n)=\Sigma m(E_n)$. 

$m$ is translation invariant.

Outer measure: $m^*(A)=\inf \Sigma l(l_n)$ taken over all countable open covers of $A$. 

Measurable set: $E$ such that for any $A$, $m^*A=m^*(A \cap E)+m^*(A\cap \bar E)$. 

(Lebesgue) measurable function: $f$ is an extended real valued function whose domain is measurable. Then, $f$ is measurable is $\forall \alpha \in \mathbb R$, the set $\{x : f(x)\ge / > / < / \le \alpha\}$ is measurable. 

A property is said to hold almost everywhere if it fails only on a set of measure 0.

Lebesgue integral

A function is called simple if it is measurable and assumes only a finite number of values.

For any simple function $\phi=\sigma a_i \chi_{A_i}$, where $A_i \in \mathcal M$, $\int_E \phi dx =\sigma a_i m(A_i \cap E)$. 

If $f$ is measurable, $\int_E f(x) dx:=\inf_{f\le \psi} \int_E \psi(x)dx$ defines the Lebesgue integral.

Theorems

Between any two $x< y \in \mathbb{R}$ there is a rational $r$.

If a sequence converges (or diverges to $\pm \infty$) its limit superior and limit superior are equal. 

The intersection of any finite union of open sets is open. 

The union/intersection of any collection of closed sets is closed.

Every open set in the real numbers is the union of a countable collection of open intervals. 

Heine-Borel: Let $F$ be a closed, bounded set of real numbers. Then each open covering of $F$ has a finite subcover. 

Let $f$ be a real valued function defined on $(-\infty, \infty)$. Then, $f$ is continuous if and only if for each open set $O$ of real numbers $f^{-1}[O]$ is an open set.

The family $\mathcal M$ of measurable sets is a $\sigma$-algebra of sets. That is, the complement of a measurable set is measurable, and the union/intersection of a countable collection of measurable sets is measurable. Moreover, every set with outer measure zero is measurable.

Every Borel set is measurable. In particular, each closed/open set is measurable.

Linear combinations and products of real valued measurable functions are measurable.

Let $f$ be defined and bounded on a measurable set $E$ with $m(E)< \infty$. In order that #\inf_{f\le \psi} \inf_e \psi(x)dx= \sup_{phi\ge \phi} \int_E \phi(x) dx# for all simple functions $\phi$ and $\psi$, it is necessary and sufficient that $f$ be measurable. 

[Stopped at BCT]

In the next part, properties of the Lebesgue integration, vector spaces. and $L^p$ spaces.