Set and Measure Theory Basics

$$\huge\textbf{Set and Measure Theory Basics} $$

1. Sequences, Sets, and Sequences of Sets

$def.$ supremum(上确界) & infimum(下确界)

Given a set $S \subseteq \mathbb R$ we define $$ \begin{align} &\sup S:=\inf \{u\mid \forall x \in S, x \leqslant u\}\\ &\inf S:=\sup \{\ell\mid \forall x \in S, x \geqslant \ell\} \end{align} $$

The well-ordering principle says that infimum and supremum always exist in $[−∞, ∞]$. For example, $\inf[a, b] = \inf[a, b) = \inf(a, b) = \inf(a, b] = a, \sup(a, ∞) = ∞, \inf ∅ = ∞, \sup ∅ = −∞, \inf ∅ = ∞$.
Given a sequence of real numbers $a_n$, $n ⩾ 1$, we say that $a_n → L$ or $lim_n→∞ a_n = L$ for some $L ∈ R$ $\text{iff}$

$$\forall ε > 0\quad\exists N = N(ε) \quad\text{s.t.} \quad\forall n ⩾ N, |a_n − L| < ε. $$

We say that $a_n → ∞$ $\text{iff}$

$$\forall M > 0 \quad\exists N = N(M) \quad\text{s.t.}\quad \forall n > N, a_n > M. $$

Note that an can diverge (发散) without going to $±∞$, e.g., by oscillating (震荡). We also have the following result.

$\textbf{Theorem 1.1.}$ (Monotone Convergence Theorem – for real numbers)$\quad$ A bounded monotone sequence of real numbers has a limit. If the sequence is increasing, the limit is the supremum, and if the sequence is decreasing, the limit is the infimum of the sequence.

Given a sequence of real numbers $a_n$, $n ⩾ 1$, we define: $$ \begin{align} \limsup _{n \rightarrow \infty} a_{n}:=\lim _{m \rightarrow \infty} \sup _{n \geqslant m} a_{n}=\inf _{m \geqslant 1} \sup _{n \geqslant m} a_{n} \\ \liminf _{n \rightarrow \infty} a_{n}:=\lim _{m \rightarrow \infty} \inf _{n \geqslant m} a_{n}=\sup _{m \geqslant 1} \inf _{n \geqslant m} a_{n} \end{align} $$

Note that, $\lim a_n$ exists $\iff \lim \sup a_n = \lim \inf a_n$. We can extend the above definition to a sequence of sets. Given a sequence of sets $A_n ⊆ \mathbb R$, $n ⩾ 1$ we define

$$\begin{array}{l}\cup_{n \geqslant 1} A_{n}:=\left\{x: \exists n \text { s.t. } x \in A_{n}\right\}=A_{1} \cup A_{2} \cup \cdots \cup A_{n} \cup \cdots \\ \cap_{n \geqslant 1} A_{n}:=\left\{x: \forall n, x \in A_{n}\right\}=A_{1} \cap A_{2} \cap \cdots \cap A_{n} \cap \cdots .\end{array} $$

$\text{De Morgan}$’s Laws say that:

$$\begin{array}{l}\left(\cup_{n \geqslant 1} A_{n}\right)^{c}=\cap_{n \geqslant 1} A_{n}^{c} \\ \left(\cap_{n \geqslant 1} A_{n}\right)^{c}=\cup_{n \geqslant 1} A_{n}^{c} .\end{array} $$

Given a sequence of sets $A_n$, $n ⩾ 1$, we now define: $$ \begin{aligned} \limsup _{n \rightarrow \infty} A_{n} & :=\bigcap_{m \geqslant 1} \bigcup_{n \geqslant m} A_{n} \\ \liminf _{n \rightarrow \infty} A_{n} & :=\bigcup_{m \geqslant 1} \bigcap_{n \geqslant m} A_{n} .\end{aligned} $$

Given a set $E$, we define the power set of $E$

$$2^E:=\{A\mid A\subseteq E\} $$

as the collection of all subsets of $E$. When $|E| < ∞$, there is a one-to-one correspondence between subsets of $E$ and binary strings of length $|E|$, $i.e.$, $\{0, 1\}^E$. Note that, $E$ is finite $\iff$ there exists a bijection $f : E → \{1, 2, . . . , n\}$ for some non-negative integer $n$. Then we say that $|E| = n$ or $E$ has $n$ elements. Similarly, $E$ is countably infinite (可列) $\iff$ there exists a bijection $f : E → \mathbb N := \{1, 2, . . .\}$. We say a set $E$ is countable if it is finite or countably infinite. For example, $\mathbb R$ is not countable but $\mathbb Q$ is countable. $2^{\mathbb N}$ is also not countable.

2. $σ$-algebras and partitions

$\textbf{Definition 1.2. }\sigma\textbf{-algebra}$ Given a set $E$, a $\sigma$-algebra $E$ on $E$ is a collection of subsets of $E$ such that $$ \left\{ \begin{aligned} & \emptyset \in \mathcal{E}\\ & A \in \mathcal{E} \Longrightarrow A^{c} \in \mathcal{E}\\ & \forall n\in \mathbb{N} :A_{n} \in \mathcal{E} \Longrightarrow \cup_{n \in \mathbb{N}} A_{n} \in \mathcal{E}\\ \end{aligned} \right. $$

The tuple $(E, \mathcal E)$ is called a measurable space. (that is the original set and its $\sigma$-algebra).

Actually, a measure on $X$ is a function that assigns a non-negative real number to subsets of $X$ this can be thought of as making precise a notion of “size” or “volume” for sets. We want the size of the union of disjoint sets to be the sum of their individual sizes, even for an infinite sequence of disjoint sets.

$\textbf{Definition 1.3. Coarse \& Fine}$ Given two $σ$-algebras $E_1 ⊆ E_2$ on E, we say that $E_1$ is coarser than $E_2$ and $E_2$ is finer than $E_1.$

Note that, $2^E$ is the biggest or finest $σ$-algebra on $E$ and $\{∅, E\}$ is the smallest or coarsest $σ$-algebra on $E$, called the trivial $σ$-algebra.

$\textbf{Definition 1.4.}$ If $\mathcal A$ is a collection of subsets of $E$, $i.e.$, $\mathcal A ⊆ 2^E$, the $σ$-algebra generated by $\mathcal A$, written as $σ(\mathcal A)$, is the smallest or coarsest $σ$-algebra containing $\mathcal A$.

$\textbf{Definition 1.5. Partition}$ A partition $\Pi$ of $E$ is a collection of subsets of $E$, namely $\Pi = \{E_i\mid i ∈ I\}$ that is

(a) $\textbf{Exhaustive}$: $\cup_{i\in I}E_i = E$

(b) $\textbf{Exclusive}$: $i\not = j\implies E_i\cap E_j=\emptyset$

Partitions can be formed as equivalence classes based on equivalence relations and vice versa.

$\textbf{Definition 1.7. Equivalence Relation}$ An equivalence relation $\sim$ on $E$ is

(a) $\textbf{Reflective}$: $\forall x\in E:x\sim x$

(b) $\textbf{Symmetric}$: $x\sim y\implies y\sim x$

© $\textbf{Transitive}$: $x \sim y\land y \sim z \implies x \sim z$.

An **equivalence class** containing $x$ is $E_x := \{y ∈ E \mid x ∼ y\}$. Note that, $\{E_x \mid x ∈ E\}$ gives a partition of $E$. Conversely, given a countable set $E$, any $σ$-algebra $\mathcal F$ on $E$ arises in the above way, $i.e.$, there exists a partition $\Pi$ on $E$ such that $\mathcal F = σ(\Pi)$. Define the relation $\sim$ on $\mathbb E$ as follows: $x ∼ y$ if $(x ∈ A \iff \forall A ∈ \mathcal F: y ∈ A)$. $∼$ is an equivalence relation. Let $\Pi := \{E_i\}_{i∈I}$ be the disjoint equivalence classes $w.r.t.$ $∼$ giving a partition of $\mathbb E$. One can prove that, $\mathcal F = σ(\Pi)$.

If $E$ is uncountable, then the above statement is not true