**Definition.** Let be a nonempty set. An algebra of sets on is a nonempty collection of subsets of that is closed under finite unions and complements; in other words, if , then ; and if , then . A -algebra is an algebra that is closed under countable unions.

**Remark.** , algebras (resp. -algebras) are also closed under finite (resp. countable) intersections. Moreover, if is an algebra, then and , for if we have and .

An algebra is a -algebra provided that it is closed under countable disjoint unions. Indeed, suppose . Set

.

Then ‘s belong to and are disjoint, and .

**Example. **If is any set, and are -algebras. If is uncountable, then

is countable or is countable

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

**Remark.** Since the intersection of any family of -algebras on is a -algebra. If follows that if is any subset of , there is a unique smallest -algebra containing , namely, the intersection of all -algebra containing . is called the -algebra generated by .

**Lemma.** If , then .

Proof. is a -algebra containing ; it therefore contains .

**Definition.** If is any metric space, or more generally any topological space, the -algebra generated by the family of open sets in (or equivalently, by the family of closed sets in ) is called the Borel -algebra on and is denoted by . Its members are called Borel sets. 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 set; a countable union of closed sets is called an set; a countable union of sets is called a set; a countable intersection of sets is called an set.

**Proposition.** is generated by each of the following:

a. the open intervals: ,

b. the closed intervals: ,

c. the half-open intervals: or ,

d. the open rays: or ,

e. the closed rays: or .

Proof. By the Lemma above, since all of the sets are Borel sets, then for all . Also, since every open set in is a countable union of open intervals, so by lemma again, . Since all open intervals lie in , then by lemma, for . For example, .

“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.