Real Analysis Review Day 11: Sigma Algebra

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May 20, 2015

Definition. Let $X$ be a nonempty set. An algebra of sets on $X$ is a nonempty collection $\mathcal A$ of subsets of $X$ that is closed under finite unions and complements; in other words, if $E_1, E_2, ..., E_n\in\mathcal A$ , then $\cup_1^n E_j\in\mathcal A$ ; and if $E\in\mathcal A$ , then $E^c\in\mathcal A$ . A $\sigma$ -algebra is an algebra that is closed under countable unions.

Remark. $\cap_j E_j=(\cap_j E_j^c)^c$ , algebras (resp. $\sigma$ -algebras) are also closed under finite (resp. countable) intersections. Moreover, if $\mathcal A$ is an algebra, then $\emptyset\in\mathcal A$ and $X\in\mathcal A$ , for if $E\in\mathcal A$ we have $\emptyset=E\cap E^c$ and $X=E\cup E^c$ .

An algebra $\mathcal A$ is a $\sigma$ -algebra provided that it is closed under countable disjoint unions. Indeed, suppose $\{E_j\}_1^\infty\subset \mathcal A$ . Set

$F_k=E_k\setminus [\cup_1^{k-1}E_j]=E_k\cap[\cup_1^{k-1}E_j]^c$ .

Then $F_k$ ‘s belong to $\mathcal A$ and are disjoint, and $\cup_1^\infty E_j=\cup_1^\infty F_k$ .

Example. If $X$ is any set, $\mathcal P(X)$ and $\{\emptyset, X\}$ are $\sigma$ -algebras. If $X$ is uncountable, then

$\mathcal A=\{E\subset X: E$ is countable or $E^c$ is countable $\}$

is a $\sigma$ -algebra, called the $\sigma$ -algebra of countable or co-countable sets.

Remark. Since the intersection of any family of $\sigma$ -algebras on $X$ is a $\sigma$ -algebra. If follows that if $\mathcal E$ is any subset of $\mathcal P(X)$ , there is a unique smallest $\sigma$ -algebra $\mathcal M(\mathcal E)$ containing $\mathcal E$ , namely, the intersection of all $\sigma$ -algebra containing $\mathcal E$ . $\mathcal M(\mathcal E)$ is called the $\sigma$ -algebra generated by $\mathcal E$ .

Lemma. If $E\subset \mathcal M(\mathcal F)$ , then $\mathcal M(\mathcal E)\subset \mathcal M(\mathcal F)$ .

Proof. $\mathcal M(\mathcal F)$ is a $\sigma$ -algebra containing $\mathcal E$ ; it therefore contains $\mathcal M(\mathcal E)$ . $\Box$

Definition. If $X$ is any metric space, or more generally any topological space, the $\sigma$ -algebra generated by the family of open sets in $X$ (or equivalently, by the family of closed sets in $X$ ) is called the Borel $\sigma$ -algebra on $X$ and is denoted by $\mathcal B_X$ . Its members are called Borel sets. $\mathcal B_x$ thus includes open sets, closed sets, countable intersections of open sets, countable unions of closed sets, and so forth.

Definition. A countable intersection of open sets is called a $G_\delta$ set; a countable union of closed sets is called an $F_\sigma$ set; a countable union of $G_\delta$ sets is called a $G_{\delta\sigma}$ set; a countable intersection of $F_\sigma$ sets is called an $F_{\sigma\delta}$ set.

Proposition. $\mathcal B_{\mathbb R}$ is generated by each of the following:

a. the open intervals: $\mathcal E_1=\{(a, b): a<b\}$ ,

b. the closed intervals: $\mathcal E_2=\{[a, b]: a<b\}$ ,

c. the half-open intervals: $\mathcal E_3=\{(a, b]: a<b\}$ or $\mathcal E_4=\{[a, b): a<b\}$ ,

d. the open rays: $\mathcal E_5=\{(a, \infty): a\in\mathbb R\}$ or $\mathcal E_6=\{(-\infty, a): a\in\mathbb R\}$ ,

e. the closed rays: $\mathcal E_7=\{[a, \infty): a\in\mathbb R\}$ or $\mathcal E_8=\{(-\infty, a]: a\in\mathbb R\}$ .

Proof. By the Lemma above, since all of the sets are Borel sets, then $\mathcal M(\mathcal E_j)\subset \mathcal B_\mathbb R$ for all $j$ . Also, since every open set in $\mathbb R$ is a countable union of open intervals, so by lemma again, $\mathcal B_{\mathbb R}\subset\mathcal M(\mathcal E_1)$ . Since all open intervals lie in $\mathcal M(\mathcal E_j)$ , then by lemma, $\mathcal B_{\mathbb R}\subset\mathcal M(\mathcal E_j)$ for $j\ge 2$ . For example, $(a, b)=\cup_1^\infty[a+n^{-1}, b-n^{-1}]\in\mathcal M(\mathcal E_2)$ . $\Box$

“What is tolerance? It is the consequence of humanity. We are all formed of frailty and error; let us pardon reciprocally each other’s folly – that is the first law of nature.” ~ Voltaire

References:

[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 21-22.

[2] Purpose Fairy’s 21-Day Happiness Challenge, http://www.jrmstart.com/wordpress/wp-content/uploads/2014/10/Free+eBook+-+PurposeFairys+21-Day+Happiness+Challenge.pdf.

Tags: real analysis, sigma algebra

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