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\},

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.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: