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.