1 Algebra, sigma-Algebra, Measurable Spaces, limsup

Definition

$X$ is a set, and $\mathcal{P}(X)$ is the set of all subsets of $X$. $\mathcal{A}\subseteq \mathcal{P}(X)$ is an algebra if

• $\varnothing ,X\in \mathcal{A}$
• $A\in \mathcal{A}$ yields that $A^c\in \mathcal{A}$
• if $\{A_i{\}}_{i=1}^n\subseteq \mathcal{A}$, then ${\cup }_{i=1}^n{A}_i\in \mathcal{A}$.

Definition

An algebra $\mathcal{A}$ is a $\sigma$-algebra if $\mathcal{A}$ is closed w.r.t. countable union, i.e. if $\{A_i{\}}_{i=1}^{\infty }\subseteq \mathcal{A}$, then ${\cup }_{i=1}^{\infty }{A}_i\in \mathcal{A}$. $(\ast )$

Remark.

• $(\ast )$ holds iff $\{A_i{\}}_{i=1}^{\infty }\subseteq A$, then ${\cap }_{i=1}^{\infty }{A}_i\in \mathcal{A}$.
• If $\mathcal{A}$ is a $\sigma$-algebra on $X$, $(X,\mathcal{A})$ is called a measure space.

Example.

• $\mathcal{A}=\{\varnothing ,X\}$, $\mathcal{P}(X)$ are $\sigma$-algebra on $X$.
• $X$ is an infinite set, then $\mathcal{A}=\{A\subseteq X:A\text{mbox}isfinite\}$ is not an algebra and ${\mathcal{A}}^{\prime }=\{A\subseteq X:A\text{mbox}isfiniteor{A}^c\text{mbox}isfinite\}$ is an algebra. Moreover, ${\mathcal{A}}^{\prime }$ is not a $\sigma$-algebra (Let $A_i=\{x_{2i+1}\}$ with $\{x_n{\}}_{n=1}^{\infty }\subseteq X$. Then $\cup A_i\notin \mathcal{A}$. ).
• $X$ is uncountable. Let $\mathcal{A}=\{A\subseteq X:A\text{mbox}iscountableor{A}^c\text{mbox}iscountable\}$. Then $\mathcal{A}$ is a $\sigma$-algebra.

Proposition

If $\{{\mathcal{A}}_i{\}}_{i\in I}$ is a family of algebras ($\sigma$-algebra), then ${\cap }_{i\in I}{\mathcal{A}}_i$ is also an algebra ($\sigma$-algebra).

Corollary

If $\mathcal{K}\subseteq \mathcal{P}(X)$, the intersection of all algebras ($\sigma$-algebras) containing $\mathcal{K}$ is the algebra ($\sigma$-algebra) generated by $\mathcal{K}$.

Definition

A Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.

Definition

Let $(X,\tau )$ be a topological space. Let $\mathcal{B}(X)$ be the $\sigma$-algebra generated by open subsets of $X$, $\mathcal{B}(X)$ is called the Borel $\sigma$-algebra on $X$.

Theorem

$\mathcal{B}(\mathbb{R})$ is generated by each of the following family of subsets:

• the collection of closed subsets of $\mathbb{R}$,
• the collection of the set of the form $(-\infty ,a]$,
• the collection of the set of the form $(a,b]$.

\begin{proof} Let $B_1,B_2,B_3$ be the $\sigma$-algebra generated by the collection of subsets in i), ii), iii), respectively. We claim that $\mathcal{B}(\mathbb{R})\supseteq B_1\supseteq B_2\supseteq B_3\supseteq \mathcal{B}(\mathbb{R})$.

Some of them are easy, and we only prove $B_2\supseteq B_3\supseteq \mathcal{B}(\mathbb{R})$. Since $(-\infty ,b]\cap (-\infty ,a{]}^c=(a,b]$, we have $B_2\supseteq B_3$. Since $(a,b)={\cup }_{n=1}^{\infty }(a,b-1/n]$, we have $B_3\supseteq \mathcal{B}(\mathbb{R})$. \end{proof}

limsup and liminf and …

Definition

We define

$$\underset{n\to \infty }{\mathrm{limsup}}{A}_n={\cap }_{n=1}^{\infty }{\cup }_{m=n}^{\infty }{A}_m,\underset{n\to \infty }{\mathrm{liminf}}{A}_n={\cup }_{n=1}^{\infty }{\cap }_{m=n}^{\infty }{A}_m.$$

Remark.

• It can be checked that ${\mathrm{limsup}}_{n\to \infty }{A}_n$ (respectively ${\mathrm{liminf}}_{n\to \infty }{A}_n)$ consists of those elements of $X$ that belong to infinitely many $A_n$ (respectively that belong to all but finitely many $A_n$ ).
• Recall that $\lim \sup a_n={\inf }_n{\sup }_{m⩾n}{a}_m$, and they are similar.

Proposition

$\mathcal{B}({\mathbb{R}}^n)$ is generated by each of the collection of sets:

• all closed subsets of ${\mathbb{R}}^n$,
• sets of the form $\{(x_1,\cdots ,x_n):x_i⩽b_i,i=1,\cdots ,n,b_i\in \mathbb{R}\}$,
• set of the form $\{(x_1,\cdots ,x_n):a_i<x_i⩽b_i,a_i,b_i\in \mathbb{R}\}$.

\begin{proof} It is similar to the case of [[#^m1q4o8|$B(\mathbb{R})$]]. \end{proof}

Definition

A $F_{\sigma }$-set is a countable union of closed sets.

A $G_{\delta }$-set is a countable intersection of open sets.