Chapter 4
Product spaces and independence
1.
Product measures
<
1
>
Definition.
Let
X
1
,...,
X
n
be sets equipped with sigma-fields
A
1
,...,
A
n
. The
set of all ordered
n
-tuples
(
x
1
,...,
x
n
)
, with
x
i
∈
X
i
for each
i
is denoted by
X
1
×
...
×
X
n
or
X
i
≤
n
X
i
. It is called the
product
of the
{
X
i
}
. A set of the form
A
1
×
...
×
A
n
= {
(
x
1
,...,
x
n
)
∈
X
1
×
...
×
X
n
:
x
i
∈
A
i
for each
i
}
,
with
A
i
∈
A
i
for each
i
, is called a
measurable rectangle
. The product sigma-field
A
1
⊗
...
⊗
A
n
on
X
1
×
...
×
X
n
is defined to be the sigma-field generated by all
measurable rectangles.
Remark.
Even if
n
equals 2 and
X
1
=
X
2
=
R
, there is is no presumption
that either
A
1
or
A
2
is an interval—a measurable rectangle might be composed of
many disjoint pieces. The symbol
⊗
in place of
×
is intended as a reminder that
A
1
⊗
A
2
consists of more than the set of all measurable rectangles
A
1
×
A
2
.
To keep the notation simple, I will mostly consider only measures on a
product of two spaces,
(
X
,
A
)
and
(
Y
,
B
)
. Sometimes I will abbreviate symbols like
M
+
(
X
×
Y
,
A
⊗
B
)
to
M
+
(
X
×
Y
)
, with the product sigma-
fi
eld assumed, or to
M
+
(
A
⊗
B
)
, with the product space assumed. Similarly,
M
bdd
(
A
⊗
B
)
will be an
abbreviation for
M
bdd
(
X
×
Y
,
A
⊗
B
)
, the vector space of all bounded, real-valued,
product measurable functions on
X
×
Y
.
Suppose
µ
is a
fi
nite measure on
A
and
ν
is a
fi
nite measure on
B
. The next
theorem, which is usually called Fubini’s Theorem, asserts existence of a
fi
nite
measure on
A
⊗
B
whose integrals can be calculated by iterated integrals with
respect to
µ
and
ν
.
Remember the notation
µ
x
h
(
x
,
y
)
for what would be written
h
(
x
,
y
)µ(
dx
)
in
traditional notation, the integral of
h
(
·
,
y
)
with respect to
µ
with
y
held
fi
xed.
<
2
>
Theorem.
For finite measures
µ
and
ν
, there is a uniquely determined finite
measure
on
A
⊗
B
for which
(i)
(
A
×
B
)
=
(µ
A
)(ν
B
)
for each measurable rectangle.
Moreover, for each
h
in
M
bdd
(
A
⊗
B
)
,