Let be an indexed collection of nonempty sets,
and
the coordinate maps. If
is a
-algebra on
for each
, the product
-algebra on
is the
-algebra generated by
.
Denote this -algebra by
. (If
we also write
or
.)
Proposition 1. If is countable then
is the
-algebra generated by
.
Proof. If , then
where
for
; on the other hand,
.
Proposition 2. Suppose that $latex\mathcal M_\alpha$ is generated by ,
. Then
is generated by
. If
is countable and
for all
,
is generated by
.
Proof. It’s easily seen that . On the other hand, for each
, the collection
is easily seen to be a
-algebra on
that contains
and hence $latex\mathcal M_\alpha$.
Proposition 3. Let be metric spaces and let
, equipped with the product metric. Then
. If the
‘s are separable, then $latex\bigotimes_1^n\mathcal B_{X_j}=\mathcal B_X$.
Proof. By the Proposition 2, is generated by the sets
,
, where
is open in
. Since these sets are open in
, Lemma implies that
. Suppose now that
is a countable dense set in
, and let
be the collection of balls in
with rational radius and center in
. Then every open set in
is a union of members of
(a countable union, since
itself is countable). Moreover, the set of points in
whose
th coordinate is in
for all
is a countable dense subset of
, and the balls of radius
in
are merely products of balls of radius
in the
‘s. It follows that
is generated by
and
is generated by
. Therefore
.
Corollary. .
Definition. An elementary family is a collection of subsets of
such that:
1) ,
2) if then
,
3) if then
is a finite disjoint union of members of
.
Proposition. If is an elementary family, the collection
of finite disjoint unions of members of
is an algebra.
Proof. If and
(
, disjoint), then
and
, where these unions are disjoint, so
and
. It now follows by induction that if
, then
; By inductive hypothesis we may assume that
are disjoint, and then
, which is a disjoint union. Suppose
and
with
disjoint members of $latex\mathcal E$, then
is closed under complements.
“The power of imagination is incredible. Often we see athletes achieving unbelievable results and wonder how they did it. One of the tools they use is visualization or mental imagery… they made the choice to create their destinies and visualized their achievements before they ultimately succeeded.” ~ George Kohlrieser
References:
[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 22-24.
[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.