More on Random Variables
Expectation :: Properties :: Pairs :: Independence :: Two R.V.S. :: Functions :: Inequalities :: Conditional :: General
A more general definition of conditional expectation
We will explore conditional expectation in terms of probability
spaces. Suppos
is a sub
field of
. A function
is ''measurable with respect to
''
if
(Any such function would be a r.v. too. But this is more restricted.)
We will now define ''conditioning on a
field.'' Suppose
is a r.v. with
and
is a sub
field
of
. Then
is any
measurable random variable
such that
If itself were measurable it would be its own conditional expectation.
Fact: A r.v.
is measurable with respect to
if and
only if there is a measurable function
such
that
.
We now define conditional expectation with respect to a
field:
Properties:

By the fact stated above, we can write
 If itself is measurable, then .
 .
 If , then

If
and
, then
Note: and are independent r.v.s iff and are independent fields.

If
is independent of
then
.

If
is
measurable then
.
So, for example, if , and for some (that is, is measurable) then