"Simple random variable" (simple
function) is a r.v. of the
form
where the
is the indicator function of the event (set)
:
and
are scalars.

Definition

The expectation (integral) of a
simple random variable
is
For any positive random variable
we define the integral as
where the
is taken over all simple positive r.v.
such that
for
.

If the
is finite then the r.v. (function of
)
is called "summable" on
.

Proposition

(Fatou lemma) If
is a sequence of non-negative random variables and
a.s. on
then

Proof

It is sufficient to show that for any simple r.v.
such that
on
we have
for any small
and sufficiently large
.

We start by writing definition of a.s. convergence according to the recipes of
the section (
Operations
on sets and logical
statements
)
up to a set of measure 0. We fix some
then
Let's introduce the
notation
By positiveness of the
and
we
have
for any simple function
,
.
The
is an increasing sequence, hence, by the proposition
(
Continuity lemma
)
Pick
such that
.
For
we
have
The last two terms are arbitrarily small and the statement holds for
sufficiently large
and any simple r.v.
.

Proposition

(Dominated convergence theorem) Let
is such that
a.s. on
,
and
.
Then
.