Real Analysis Review Day 11: Sigma Algebra

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

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

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.